JEE Main & Advanced Mathematics Functions Question Bank

done Limits Total Questions - 227

• question_answer1)
  If \[f(x)=\left\{ \begin{align}   & x\sin \frac{1}{x},\ \ \ \ \ x\ne 0 \\  & \ \ \ \ \ \ 0,\ \ \ \ \ x=0 \\ \end{align} \right.\],  then \[\underset{x\to 0}{\mathop{\lim }}\,f(x)=\]                 [IIT 1988; MNR 1988; SCRA 1996; UPSEAT 2000, 01]
  A)
                  1 done clear
  B)
                  0 done clear
  C)
                  ?1 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer2)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{x}^{3}}\cot x}{1-\cos x}=\] [AI CBSE 1988; DSSE 1988]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  2 done clear
  D)
                  ?2 done clear
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• question_answer3)
                                                                                                                                                                      \[\underset{x\to 0}{\mathop{\lim }}\,\frac{x({{e}^{x}}-1)}{1-\cos x}=\]
  A)
                  0 done clear
  B)
                  \[\infty \] done clear
  C)
                  ?2 done clear
  D)
                  2 done clear
  View Solution play_arrow

• question_answer4)
  \[\underset{x\to 1}{\mathop{\lim }}\,\frac{1}{|1-x|}=\]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  2 done clear
  D)
                  \[\infty \] done clear
  View Solution play_arrow

• question_answer5)
   \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{n{{(2n+1)}^{2}}}{(n+2)({{n}^{2}}+3n-1)}=\]
  A)
                  0 done clear
  B)
                  2 done clear
  C)
                  4 done clear
  D)
                  \[\infty \] done clear
  View Solution play_arrow

• question_answer6)
  \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{\sqrt{n}}{\sqrt{n}+\sqrt{n+1}}=\]
  A)
                  1 done clear
  B)
                  1/2 done clear
  C)
                  0 done clear
  D)
                  \[\infty \] done clear
  View Solution play_arrow

• question_answer7)
  \[\underset{x\to a}{\mathop{\lim }}\,\frac{\sqrt{3x-a}-\sqrt{x+a}}{x-a}=\]
  A)
                  \[\sqrt{2}a\] done clear
  B)
                  \[1/\sqrt{2a}\] done clear
  C)
                  2a done clear
  D)
                  \[1/2a\] done clear
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• question_answer8)
  If \[f(x)=\left\{ \begin{align}   & \,\,\,\,\,\,\,x,\ \text{when }0\le x\le 1 \\  & 2-x,\ \text{when }1<x\le 2 \\ \end{align} \right.\],  then \[\underset{x\to 1}{\mathop{\lim }}\,f(x)=\]
  A)
                  1 done clear
  B)
                  2 done clear
  C)
                  0 done clear
  D)
                  Does not exist done clear
  View Solution play_arrow

• question_answer9)
  \[\underset{x\to 1}{\mathop{\lim }}\,\frac{\log x}{x-1}=\] [RPET 1996; MP PET 1996; Pb. CET 2002]
  A)
                  1 done clear
  B)
                  ?1 done clear
  C)
                  0 done clear
  D)
                  \[\infty \] done clear
  View Solution play_arrow

• question_answer10)
  If \[\underset{x\to 2}{\mathop{\lim }}\,\frac{{{x}^{n}}-{{2}^{n}}}{x-2}=80\], where n is a positive integer, then \[n=\]                
  A)
                  3 done clear
  B)
                  5 done clear
  C)
                  2 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer11)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos 2x}{x}=\] [MNR 1983]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  2 done clear
  D)
                  4 done clear
  View Solution play_arrow

• question_answer12)
  \[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{2}{x} \right)}^{x}}=\]
  A)
                  e done clear
  B)
                  \[\frac{1}{e}\] done clear
  C)
                  \[{{e}^{2}}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer13)
  \[\underset{x\to 1}{\mathop{\lim }}\,\frac{(2x-3)(\sqrt{x}-1)}{2{{x}^{2}}+x-3}=\]                                             [IIT 1977]
  A)
                  ?1/10 done clear
  B)
                  1/10 done clear
  C)
                  ?1/8 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer14)
  If \[\underset{x\to 0}{\mathop{\lim }}\,kx\,\text{cosec}\,x=\underset{x\to 0}{\mathop{\lim }}\,x\,\text{cosec}\ kx\], then \[k=\]
  A)
                  1 done clear
  B)
                  ?1 done clear
  C)
                  \[\pm 1\] done clear
  D)
                  \[\pm \,2\] done clear
  View Solution play_arrow

• question_answer15)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{1/x}}-1}{{{e}^{1/x}}+1}=\]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  ?1 done clear
  D)
                  Does not exist done clear
  View Solution play_arrow

• question_answer16)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\log \cos x}{x}=\]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  \[\infty \] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer17)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin 2x}{x}=\]             [MNR 1990; UPSEAT 2000]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  ½ done clear
  D)
                  2 done clear
  View Solution play_arrow

• question_answer18)
  If \[f(9)=9\], \[f'(9)=4\], then \[\underset{x\to 9}{\mathop{\lim }}\,\frac{\sqrt{f(x)}-3}{\sqrt{x}-3}=\]                 [IIT 1988; Karnataka CET 1999]
  A)
                  2 done clear
  B)
                  4 done clear
  C)
                  ?2 done clear
  D)
                  ?4 done clear
  View Solution play_arrow

• question_answer19)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{|x|}{x}=\]   [Roorkee 1982; UPSEAT 2001]
  A)
                  1 done clear
  B)
                  ?1 done clear
  C)
                  0 done clear
  D)
                  Does not exist done clear
  View Solution play_arrow

• question_answer20)
  \[\underset{h\to 0}{\mathop{\lim }}\,\frac{\sqrt{x+h}-\sqrt{x}}{h}=\] [Roorkee 1983]
  A)
                  \[\frac{1}{2\sqrt{x}}\] done clear
  B)
                  \[\frac{1}{\sqrt{x}}\] done clear
  C)
                  \[2\sqrt{x}\] done clear
  D)
                  \[\sqrt{x}\] done clear
  View Solution play_arrow

• question_answer21)
   \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{2}^{x}}-1}{{{(1+x)}^{1/2}}-1}=\]    [IIT 1983; Karnataka CET 1999]
  A)
                  \[\log 2\] done clear
  B)
                  \[\log 4\] done clear
  C)
                  \[\log \sqrt{2}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer22)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos mx}{1-\cos nx}=\] [Kerala (Engg.)2002]
  A)
                  \[m/n\] done clear
  B)
                  \[n/m\] done clear
  C)
                  \[\frac{{{m}^{2}}}{{{n}^{2}}}\] done clear
  D)
                  \[\frac{{{n}^{2}}}{{{m}^{2}}}\] done clear
  View Solution play_arrow

• question_answer23)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{\sin x}}-1}{x}=\]
  A)
                  1 done clear
  B)
                  e done clear
  C)
                  1/e done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer24)
  \[\underset{x\to \infty }{\mathop{\lim }}\,\sqrt{x}(\sqrt{x+5}-\sqrt{x})=\]
  A)
                  5 done clear
  B)
                  3 done clear
  C)
                  5/2 done clear
  D)
                  3/2 done clear
  View Solution play_arrow

• question_answer25)
  \[\underset{x\to 1}{\mathop{\lim }}\,\frac{x-1}{2{{x}^{2}}-7x+5}=\] [IIT 1976]
  A)
                  1/3 done clear
  B)
                  1/11 done clear
  C)
                  ?1/3 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer26)
  \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{\sin x}{x}=\] [IIT 1975; MP PET 2004]
  A)
                  1 done clear
  B)
                  0 done clear
  C)
                  Does not exist done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer27)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{1+\sin x}-\sqrt{1-\sin x}}{x}=\]
  A)
                  ?1 done clear
  B)
                  1 done clear
  C)
                  2 done clear
  D)
                  ?2 done clear
  View Solution play_arrow

• question_answer28)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{2{{\sin }^{2}}3x}{{{x}^{2}}}=\] [Roorkee 1982; DCE 1999]
  A)
                  6 done clear
  B)
                  9 done clear
  C)
                  18 done clear
  D)
                  3 done clear
  View Solution play_arrow

• question_answer29)
  \[\underset{\alpha \to \pi /4}{\mathop{\lim }}\,\frac{\sin \alpha -\cos \alpha }{\alpha -\frac{\pi }{4}}=\] [IIT 1977]
  A)
                  \[\sqrt{2}\] done clear
  B)
                  \[1/\sqrt{2}\] done clear
  C)
                  1 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer30)
  \[\underset{x\to \pi /2}{\mathop{\lim }}\,\tan x\log \sin x=\] [MNR 1989]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  ?1 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer31)
  If n is an integer, then \[\underset{x\to n+0}{\mathop{\lim }}\,(x-[n])=\]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  ?1 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer32)
  \[\underset{x\to \pi /2}{\mathop{\lim }}\,(\sec \theta -\tan \theta )=\] [IIT 1976; AMU 1999]
  A)
                  0 done clear
  B)
                  1/2 done clear
  C)
                  2 done clear
  D)
                  \[\infty \] done clear
  View Solution play_arrow

• question_answer33)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\tan 2x-x}{3x-\sin x}=\] [IIT 1971]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  ½ done clear
  D)
                  1/3 done clear
  View Solution play_arrow

• question_answer34)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{x}{|x|+{{x}^{2}}}=\]
  A)
                  1 done clear
  B)
                  ?1 done clear
  C)
                  0 done clear
  D)
                  Does not exist done clear
  View Solution play_arrow

• question_answer35)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin ax}{\sin bx}=\]
  A)
                  \[a/b\] done clear
  B)
                  \[b/a\] done clear
  C)
                  1 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer36)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin {{x}^{o}}}{x}=\]
  A)
                  1 done clear
  B)
                  \[\pi /180\] done clear
  C)
                  Does not exist done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer37)
  \[\underset{x\to a}{\mathop{\lim }}\,\frac{{{x}^{2}}-{{a}^{2}}}{x-a}=\] [RPET 1995]
  A)
                  4a done clear
  B)
                  1 done clear
  C)
                  2a done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer38)
  \[\underset{x\to a}{\mathop{\lim }}\,\frac{{{(x+2)}^{5/3}}-{{(a+2)}^{5/3}}}{x-a}=\] [AI CBSE 1991]
  A)
                  \[\frac{5}{3}{{(a+2)}^{2/3}}\] done clear
  B)
                  \[\frac{5}{3}{{(a+2)}^{5/3}}\] done clear
  C)
                  \[\frac{5}{3}{{a}^{2/3}}\] done clear
  D)
                  \[\frac{5}{3}{{a}^{5/3}}\] done clear
  View Solution play_arrow

• question_answer39)
  If \[f(x)=\left\{ \begin{matrix}    \frac{2}{5-x}, & \text{when }x<3  \\    5-x, & \text{when }x>3  \\ \end{matrix} \right.\],  then
  A)
                  \[\underset{x\to 3+}{\mathop{\lim }}\,f(x)=0\] done clear
  B)
                  \[\underset{x\to 3-}{\mathop{\lim }}\,f(x)=0\] done clear
  C)
                  \[\underset{x\to 3+}{\mathop{\lim }}\,f(x)\ne \underset{x\to 3-}{\mathop{\lim }}\,f(x)\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer40)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\cos ax-\cos bx}{{{x}^{2}}}=\] [AI CBSE 1988]
  A)
                  \[\frac{{{a}^{2}}-{{b}^{2}}}{2}\] done clear
  B)
                  \[\frac{{{b}^{2}}-{{a}^{2}}}{2}\] done clear
  C)
                  \[{{a}^{2}}-{{b}^{2}}\]    done clear
  D)
                  \[{{b}^{2}}-{{a}^{2}}\] done clear
  View Solution play_arrow

• question_answer41)
  \[\underset{x\to \pi /6}{\mathop{\lim }}\,\frac{{{\cot }^{2}}\theta -3}{\text{cosec}\theta -2}=\]
  A)
                  2 done clear
  B)
                  4 done clear
  C)
                  6 done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer42)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{(1+x)}^{5}}-1}{{{(1+x)}^{3}}-1}=\]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  5/3 done clear
  D)
                  3/5 done clear
  View Solution play_arrow

• question_answer43)
  If \[\underset{x\to a}{\mathop{\lim }}\,\frac{{{x}^{9}}+{{a}^{9}}}{x+a}=9\], then \[a=\]
  A)
                  \[{{9}^{1/8}}\] done clear
  B)
                  \[\pm 2\] done clear
  C)
                  \[\pm 3\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer44)
  \[\underset{x\to 0+}{\mathop{\lim }}\,\frac{x{{e}^{1/x}}}{1+{{e}^{1/x}}}=\]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  \[\infty \] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer45)
  \[\underset{x\to 1}{\mathop{\lim }}\,[x]=\]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  Does not exist done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer46)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin 2x+\sin 6x}{\sin 5x-\sin 3x}=\] [AI CBSE 1988; AISSE 1988]
  A)
                  ½ done clear
  B)
                  1/4 done clear
  C)
                  2 done clear
  D)
                  4 done clear
  View Solution play_arrow

• question_answer47)
  The value of \[\underset{\theta \to 0}{\mathop{\lim }}\,\left( \frac{\sin \frac{\theta }{4}}{\theta } \right)\] is                                     [MP PET 1993]
  A)
                  0 done clear
  B)
                  1/4 done clear
  C)
                  1 done clear
  D)
                  Not in existence done clear
  View Solution play_arrow

• question_answer48)
  The value of \[\underset{x\to \infty }{\mathop{\lim }}\,\left( \frac{{{x}^{2}}+bx+4}{{{x}^{2}}+ax+5} \right)\] is [MP PET 1993]
  A)
                  b/a done clear
  B)
                  1 done clear
  C)
                  0 done clear
  D)
                  4/5 done clear
  View Solution play_arrow

• question_answer49)
  If \[f(r)=\pi {{r}^{2}}\], then \[\underset{h\to 0}{\mathop{\lim }}\,\frac{f(r+h)-f(r)}{h}=\]
  A)
                  \[\pi {{r}^{2}}\] done clear
  B)
                  \[2\pi r\] done clear
  C)
                  \[2\pi \] done clear
  D)
                  \[2\pi {{r}^{2}}\] done clear
  View Solution play_arrow

• question_answer50)
  \[\underset{x\to 0}{\mathop{\lim }}\,x\log (\sin x)=\]
  A)
                  ?1 done clear
  B)
                  \[{{\log }_{e}}1\] done clear
  C)
                  1 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer51)
  \[\underset{x\to 0}{\mathop{\lim }}\,\left( \frac{{{a}^{x}}-{{b}^{x}}}{x} \right)=\]            [EAMCET 1988; RPET 1995]
  A)
                  \[\log \left( \frac{b}{a} \right)\] done clear
  B)
                  \[\log \left( \frac{a}{b} \right)\] done clear
  C)
                  \[\frac{a}{b}\] done clear
  D)
                  \[\log {{a}^{b}}\] done clear
  View Solution play_arrow

• question_answer52)
  \[\underset{x\to 0}{\mathop{\lim }}\,\left\{ \frac{\sin x-x+\frac{{{x}^{3}}}{6}}{{{x}^{5}}} \right\}=\] [MNR 1985]
  A)
                  1/120 done clear
  B)
                  ?1/120 done clear
  C)
                  1/20 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer53)
  \[\underset{x\to \infty }{\mathop{\lim }}\,[x({{a}^{1/x}}-1)]\],\[(a>1)=\]
  A)
                  \[\log x\] done clear
  B)
                  1 done clear
  C)
                  0 done clear
  D)
                  \[-\log \frac{1}{a}\] done clear
  View Solution play_arrow

• question_answer54)
  \[\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{1}{x}-\frac{\log (1+x)}{{{x}^{2}}} \right]\]=
  A)
                  ½ done clear
  B)
                  ?1/2 done clear
  C)
                  1 done clear
  D)
                  ?1 done clear
  View Solution play_arrow

• question_answer55)
  \[\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{\Sigma {{n}^{2}}}{{{n}^{3}}} \right]=\]      [AMU 1999; RPET 1999, 2002]
  A)
                  \[-\frac{1}{6}\] done clear
  B)
                  \[\frac{1}{6}\] done clear
  C)
                  \[\frac{1}{3}\] done clear
  D)
                  \[-\frac{1}{3}\] done clear
  View Solution play_arrow

• question_answer56)
  If \[f(a)=2,\ f'(a)=1,\ g(a)=-1;\ g'(a)=2\], then \[\underset{x\to a}{\mathop{\lim }}\,\frac{g(x)f(a)-g(a)f(x)}{x-a}=\] [DCE 1999; Karnataka CET 1999; MP PET 1995; Pb. CET 2004]
  A)
                  3 done clear
  B)
                  5 done clear
  C)
                  0 done clear
  D)
                  ?3 done clear
  View Solution play_arrow

• question_answer57)
  \[\underset{x\to \alpha }{\mathop{\lim }}\,\frac{\sin x-\sin \alpha }{x-\alpha }=\]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  \[\sin \alpha \] done clear
  D)
                  \[\cos \alpha \] done clear
  View Solution play_arrow

• question_answer58)
  \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{\sqrt{{{x}^{2}}+{{a}^{2}}}-\sqrt{{{x}^{2}}+{{b}^{2}}}}{\sqrt{{{x}^{2}}+{{c}^{2}}}-\sqrt{{{x}^{2}}+{{d}^{2}}}}=\]
  A)
                  \[\frac{{{a}^{2}}-{{b}^{2}}}{{{c}^{2}}-{{d}^{2}}}\] done clear
  B)
                  \[\frac{{{a}^{2}}+{{b}^{2}}}{{{c}^{2}}-{{d}^{2}}}\] done clear
  C)
                  \[\frac{{{a}^{2}}+{{b}^{2}}}{{{c}^{2}}+{{d}^{2}}}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer59)
  \[\underset{x\to \pi /2}{\mathop{\lim }}\,\frac{2x-\pi }{\cos x}=\] [IIT 1973]
  A)
                  2 done clear
  B)
                  1 done clear
  C)
                  ?2 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer60)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin x-x}{{{x}^{3}}}=\] [MNR 1980, 86]
  A)
                  \[\frac{1}{3}\] done clear
  B)
                  \[-\frac{1}{3}\] done clear
  C)
                  \[\frac{1}{6}\] done clear
  D)
                  \[-\frac{1}{6}\] done clear
  View Solution play_arrow

• question_answer61)
  \[\underset{h\to 0}{\mathop{\lim }}\,\frac{{{(a+h)}^{2}}\sin (a+h)-{{a}^{2}}\sin a}{h}=\] [IIT 1989]
  A)
                  \[a\cos a+{{a}^{2}}\sin a\] done clear
  B)
                  \[a\sin a+{{a}^{2}}\cos a\] done clear
  C)
                  \[2a\sin a+{{a}^{2}}\cos a\] done clear
  D)
                  \[2a\cos a+{{a}^{2}}\sin a\] done clear
  View Solution play_arrow

• question_answer62)
  \[\underset{x\to 3}{\mathop{\lim }}\,\left\{ \frac{x-3}{\sqrt{x-2}-\sqrt{4-x}} \right\}=\] [MNR 1991]
  A)
                  1 done clear
  B)
                  2 done clear
  C)
                  ?1 done clear
  D)
                  ?2 done clear
  View Solution play_arrow

• question_answer63)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{x\cos x-\sin x}{{{x}^{2}}\sin x}=\] [MNR 1984,86]
  A)
                  \[\frac{1}{3}\] done clear
  B)
                  \[-\frac{1}{3}\] done clear
  C)
                  1 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer64)
  \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{(x-1)(2x+3)}{{{x}^{2}}}=\]
  A)
                  1 done clear
  B)
                  ?1 done clear
  C)
                  2 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer65)
  \[\underset{x\to \infty }{\mathop{\lim }}\,\left[ \frac{{{1}^{3}}+{{2}^{3}}+{{3}^{3}}+.......+{{n}^{3}}}{{{n}^{4}}} \right]=\]
  A)
                  \[\frac{1}{2}\] done clear
  B)
                  \[\frac{1}{3}\] done clear
  C)
                  \[\frac{1}{4}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer66)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{y}^{2}}}{x}=........\], where \[{{y}^{2}}=ax+b{{x}^{2}}+c{{x}^{3}}\]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  a done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer67)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{(1+x)}^{1/2}}-{{(1-x)}^{1/2}}}{x}=\] [Roorkee 1979; RPET 1996]
  A)
                  0 done clear
  B)
                  1/2 done clear
  C)
                  1 done clear
  D)
                  ?1 done clear
  View Solution play_arrow

• question_answer68)
  \[\underset{x\to 1}{\mathop{\lim }}\,\frac{{{x}^{3}}-1}{{{x}^{2}}+5x-6}=\]
  A)
                  0 done clear
  B)
                  \[\frac{3}{7}\] done clear
  C)
                  \[\frac{1}{2}\] done clear
  D)
                  \[-\frac{1}{6}\] done clear
  View Solution play_arrow

• question_answer69)
  \[\underset{x\to a}{\mathop{\lim }}\,\frac{\sqrt{a+2x}-\sqrt{3x}}{\sqrt{3a+x}-2\sqrt{x}}=\] [IIT 1978; Kurukshetra CEE 1996]
  A)
                  \[\frac{1}{\sqrt{3}}\] done clear
  B)
                  \[\frac{2}{3\sqrt{3}}\] done clear
  C)
                  \[\frac{2}{\sqrt{3}}\] done clear
  D)
                  \[\frac{2}{3}\] done clear
  View Solution play_arrow

• question_answer70)
  \[\underset{x\to 1}{\mathop{\lim }}\,\frac{1-{{x}^{-1/3}}}{1-{{x}^{-2/3}}}=\] [AI CBSE 1991]
  A)
                  \[\frac{1}{3}\] done clear
  B)
                  \[\frac{1}{2}\] done clear
  C)
                  \[\frac{2}{3}\] done clear
  D)
                  \[-\frac{2}{3}\] done clear
  View Solution play_arrow

• question_answer71)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{(1+x)}^{n}}-1}{x}=\] [Kurukshetra CEE 2002]
  A)
                  n done clear
  B)
                  1 done clear
  C)
                  ?1 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer72)
  \[\underset{x\to 0}{\mathop{\lim }}\,\left( \frac{\tan 3x}{x}+\cos x \right)=\]
  A)
                  3 done clear
  B)
                  1 done clear
  C)
                  4 done clear
  D)
                  2 done clear
  View Solution play_arrow

• question_answer73)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{1+x}-\sqrt{1-x}}{{{\sin }^{-1}}x}=\] [AI CBSE 1989, 90; DSSE 1989]
  A)
                  2 done clear
  B)
                  1 done clear
  C)
                  ?1 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer74)
  \[\underset{y\to 0}{\mathop{\lim }}\,\frac{(x+y)\sec (x+y)-x\sec x}{y}=\] [AI CBSE 1990]
  A)
                  \[\sec x(x\tan x+1)\] done clear
  B)
                  \[x\tan x+\sec x\] done clear
  C)
                  \[x\sec x+\tan x\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer75)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{x{{.2}^{x}}-x}{1-\cos x}=\]    [IIT 1980; BIT Ranchi 1983; RPET 2001]
  A)
                  0 done clear
  B)
                  \[\log 4\] done clear
  C)
                  \[\log 2\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer76)
  \[\underset{\theta \to 0}{\mathop{\lim }}\,\frac{1-\cos \theta }{{{\theta }^{2}}}=\] [AI CBSE 1981, 91; DSSE 1981, 83]
  A)
                  1 done clear
  B)
                  2 done clear
  C)
                  \[\frac{1}{2}\] done clear
  D)
                  \[\frac{1}{4}\] done clear
  View Solution play_arrow

• question_answer77)
   \[\underset{\theta \to 0}{\mathop{\lim }}\,\frac{\sin 3\theta -\sin \theta }{\sin \theta }=\] [AI CBSE 1984; DSSE 1984]
  A)
                  1 done clear
  B)
                  2 done clear
  C)
                  1/3 done clear
  D)
                  3/2 done clear
  View Solution play_arrow

• question_answer78)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\tan x-\sin x}{{{x}^{3}}}=\] [IIT 1974; AI CBSE 1986, 90; AISSE 1983, 86, 90; RPET 2000]
  A)
                  \[\frac{1}{2}\] done clear
  B)
                  \[-\frac{1}{2}\] done clear
  C)
                  \[\frac{2}{3}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer79)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos x}{x}=\] [AI CBSE 1987; AISSE 1987]
  A)
                  0 done clear
  B)
                  \[\frac{1}{2}\] done clear
  C)
                  \[\frac{1}{3}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer80)
  \[\underset{x\to 1}{\mathop{\lim }}\,\frac{1-\sqrt{x}}{{{({{\cos }^{-1}}x)}^{2}}}=\] [AI CBSE 1990]
  A)
                  1 done clear
  B)
                  \[\frac{1}{2}\] done clear
  C)
                  \[\frac{1}{4}\] done clear
  D)
                     Put \[{{\cos }^{-1}}x=y\] and \[x\to 1\,\Rightarrow \,\,y\to 0.\]            \[\underset{x\to 1}{\mathop{\lim }}\,\,\frac{1-\sqrt{x}}{{{({{\cos }^{-1}}x)}^{2}}}=\underset{y\to 0}{\mathop{\lim }}\,\,\frac{1-\sqrt{\cos y}}{{{y}^{2}}}\]            Now rationalizing it, we get \[\underset{y\to 0}{\mathop{\lim }}\,\,\frac{(1-\cos y)}{{{y}^{2}}(1+\sqrt{\cos y})}\]                                 \[=\underset{y\to 0}{\mathop{\lim }}\,\,\frac{1-\cos y}{{{y}^{2}}}\,.\,\underset{y\to 0}{\mathop{\lim }}\,\,\frac{1}{1+\sqrt{\cos y}}=\frac{1}{2}\times \frac{1}{2}=\frac{1}{4}.\] done clear
  View Solution play_arrow

• question_answer81)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{x}^{2}}-\tan 2x}{\tan x}=\]                                               [AI CBSE 1990]
  A)
                  2 done clear
  B)
                  ?2 done clear
  C)
                  0 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer82)
  \[\underset{\theta \to 0}{\mathop{\lim }}\,\frac{5\theta \cos \theta -2\sin \theta }{3\theta +\tan \theta }=\] [AI CBSE 1988]
  A)
                  \[\frac{3}{4}\] done clear
  B)
                  \[-\frac{3}{4}\] done clear
  C)
                  0 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer83)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin (2+x)-\sin (2-x)}{x}=\]                 [AI CBSE 1983; AISSE 1982, 83]
  A)
                  \[\sin 2\] done clear
  B)
                  \[2\sin 2\] done clear
  C)
                  \[2\cos 2\] done clear
  D)
                  2 done clear
  View Solution play_arrow

• question_answer84)
  \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{2{{x}^{2}}-3x+1}{{{x}^{2}}-1}=\]
  A)
                  1 done clear
  B)
                  2 done clear
  C)
                  ?2 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer85)
  \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{3{{x}^{2}}+2x-1}{2{{x}^{2}}-3x-3}=\]
  A)
                  1 done clear
  B)
                  3 done clear
  C)
                  \[\frac{3}{2}\] done clear
  D)
                  \[-\frac{3}{2}\] done clear
  View Solution play_arrow

• question_answer86)
  \[\underset{x\to 2}{\mathop{\lim }}\,\frac{|x-2|}{x-2}=\] [AI CBSE 1985]
  A)
                  1 done clear
  B)
                  ?1 done clear
  C)
                  Does not exist done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer87)
  \[\underset{x\to \pi /4}{\mathop{\lim }}\,\frac{\sqrt{2}\cos x-1}{\cot x-1}=\] [BIT Ranchi 1989; IIT 1990]
  A)
                  \[\frac{1}{\sqrt{2}}\] done clear
  B)
                  \[\frac{1}{2}\] done clear
  C)
                  \[\frac{1}{2\sqrt{2}}\] done clear
  D)
                  1 done clear
  View Solution play_arrow

• question_answer88)
  \[\underset{x\to a}{\mathop{\lim }}\,\frac{\cos x-\cos a}{\cos x-\cot a}=\] [BIT Ranchi 1987]
  A)
                  \[\frac{1}{2}{{\sin }^{3}}a\] done clear
  B)
                  \[\frac{1}{2}\text{cose}{{\text{c}}^{2}}a\] done clear
  C)
                  \[{{\sin }^{3}}a\] done clear
  D)
                  \[\text{cose}{{\text{c}}^{3}}a\] done clear
  View Solution play_arrow

• question_answer89)
  \[\underset{h\to 0}{\mathop{\lim }}\,\frac{2\left[ \sqrt{3}\sin \left( \frac{\pi }{6}+h \right)-\cos \left( \frac{\pi }{6}+h \right) \right]}{\sqrt{3}h(\sqrt{3}\cos h-\sin h)}=\] [BIT Ranchi 1987]
  A)
                  \[-\frac{2}{3}\] done clear
  B)
                  \[-\frac{3}{4}\] done clear
  C)
                  \[-2\sqrt{3}\] done clear
  D)
                  \[\frac{4}{3}\] done clear
  View Solution play_arrow

• question_answer90)
  \[\underset{x\to 0}{\mathop{\lim }}\,{{x}^{x}}=\]  [Roorkee 1990]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  e done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer91)
  \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{{{(2x+1)}^{40}}{{(4x-1)}^{5}}}{{{(2x+3)}^{45}}}=\] [IIT 1990]
  A)
                  16 done clear
  B)
                  24 done clear
  C)
                  32 done clear
  D)
                  8 done clear
  View Solution play_arrow

• question_answer92)
  \[\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{x}{{{\tan }^{-1}}2x} \right]=\]  [IIT 1992; RPET 2001]
  A)
                  0 done clear
  B)
                  \[\frac{1}{2}\] done clear
  C)
                  1 done clear
  D)
                  \[\infty \] done clear
  View Solution play_arrow

• question_answer93)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos x}{{{\sin }^{2}}x}=\] [DSSE 1987]
  A)
                  \[\frac{1}{2}\] done clear
  B)
                  \[-\frac{1}{2}\] done clear
  C)
                  2 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer94)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin 3x+\sin x}{x}\] = [AISSE 1986]
  A)
                  \[\frac{1}{3}\] done clear
  B)
                  3 done clear
  C)
                  4 done clear
  D)
                  \[\frac{1}{4}\] done clear
  View Solution play_arrow

• question_answer95)
  \[\underset{x\to \pi /2}{\mathop{\lim }}\,\frac{1+\cos 2x}{{{(\pi -2x)}^{2}}}=\] [DSSE 1986; AI CBSE 1986]
  A)
                  \[1\] done clear
  B)
                  \[2\] done clear
  C)
                  3 done clear
  D)
                  \[\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer96)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{1-\cos 6x}{x}=\] [DSSE 1982]
  A)
                  0 done clear
  B)
                  6 done clear
  C)
                  \[\frac{1}{3}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer97)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin mx}{\tan nx}=\] [DSSE 1987]
  A)
                  \[\frac{n}{m}\] done clear
  B)
                  \[\frac{m}{n}\] done clear
  C)
                  \[mn\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer98)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{3\sin x-\sin 3x}{{{x}^{3}}}=\] [AISSE 1985]
  A)
                  4 done clear
  B)
                  ?4 done clear
  C)
                  \[\frac{1}{4}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer99)
  \[\underset{x\to 0}{\mathop{\text{lim}}}\,\frac{{{x}^{3}}}{\sin {{x}^{2}}}=\] [AISSE 1984; AI CBSE 1984]
  A)
                  0 done clear
  B)
                  \[\frac{1}{3}\] done clear
  C)
                  3 done clear
  D)
                  \[\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer100)
  If \[f(x)=\left\{ \begin{align}   & x,\ \ \ \,\,\text{when}\ x>1 \\  & {{x}^{2}},\,\,\,\text{when}\,\,x<1 \\ \end{align} \right.\], then \[\underset{x\to 1}{\mathop{\lim }}\,f(x)=\] [MP PET 1987]
  A)
                  \[{{x}^{2}}\] done clear
  B)
                  \[x\] done clear
  C)
                  \[-1\] done clear
  D)
                  1 done clear
  View Solution play_arrow

• question_answer101)
  \[\underset{x\to \pi /2}{\mathop{\lim }}\,\frac{\tan 3x}{x}=\] [MP PET 1987]
  A)
                  \[\infty \] done clear
  B)
                  \[3\] done clear
  C)
                  \[\frac{1}{3}\] done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer102)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{3+x}-\sqrt{3-x}}{x}=\] [MP PET 1987]
  A)
                  ?1 done clear
  B)
                  0 done clear
  C)
                  \[\sqrt{3}\] done clear
  D)
                  \[\frac{1}{\sqrt{3}}\] done clear
  View Solution play_arrow

• question_answer103)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{{{x}^{2}}}}-\cos x}{{{x}^{2}}}=\] [IIT Screening]
  A)
                  \[\frac{3}{2}\] done clear
  B)
                  \[-\frac{1}{2}\] done clear
  C)
                  1 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer104)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\log (a+x)-\log a}{x}+k\underset{x\to e}{\mathop{\lim }}\,\frac{\log x-1}{x-e}=1,\]then                              [IIT Screening]
  A)
                  \[k=e\left( 1-\frac{1}{a} \right)\] done clear
  B)
                  \[k=e(1+a)\] done clear
  C)
                  \[k=e(2-a)\]       done clear
  D)
                  The equality is not possible done clear
  View Solution play_arrow

• question_answer105)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{\frac{1}{2}(1-\cos 2x)}}{x}=\]  [IIT 1991; AIEEE 2002; RPET 2001, 02]
  A)
                  1 done clear
  B)
                  ?1 done clear
  C)
                  0 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer106)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{x}{\tan x}\]is equal to [MNR 1995]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  4 done clear
  D)
                  Not defined done clear
  View Solution play_arrow

• question_answer107)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{\alpha \ x}}-{{e}^{\beta \ x}}}{x}=\]     [MP PET 1994; DCE 2005]
  A)
                  \[\alpha +\beta \] done clear
  B)
                  \[\frac{1}{\alpha }+\beta \] done clear
  C)
                  \[{{\alpha }^{2}}-{{\beta }^{2}}\] done clear
  D)
                  \[\alpha -\beta \] done clear
  View Solution play_arrow

• question_answer108)
  \[\underset{x\to a}{\mathop{\lim }}\,\frac{({{x}^{-1}}-{{a}^{-1}})}{x-a}=\] [MP PET 1994]
  A)
                  \[{{x}_{n+1}}=\sqrt{2+{{x}_{n}}},\ n\ge 1,\ \] done clear
  B)
                  \[\frac{-1}{a}\] done clear
  C)
                  \[\frac{1}{{{a}^{2}}}\] done clear
  D)
                  \[\frac{-1}{{{a}^{2}}}\] done clear
  View Solution play_arrow

• question_answer109)
  \[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( \frac{x+2}{x+1} \right)}^{x+3}}\] is [MNR 1994]
  A)
                  \[1\] done clear
  B)
                  \[e\] done clear
  C)
                  \[{{e}^{2}}\] done clear
  D)
                  \[{{e}^{3}}\] done clear
  View Solution play_arrow

• question_answer110)
  \[\underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,|(1-\sin x)\tan x\] is
  A)
                  \[\frac{\pi }{2}\] done clear
  B)
                  1 done clear
  C)
                  0 done clear
  D)
                  \[\infty \] done clear
  View Solution play_arrow

• question_answer111)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin x}{x}\]is equal to [RPET 1995]
  A)
                  1 done clear
  B)
                  0 done clear
  C)
                  ?1 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer112)
  \[\underset{x\to \infty }{\mathop{\lim }}\,(\sqrt{{{x}^{2}}+1}-x)\]is equal to [RPET 1995]
  A)
                  1 done clear
  B)
                  ?1 done clear
  C)
                  0 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer113)
  \[y\]exists, if [RPET 1995]
  A)
                  \[{{x}_{n+1}}=\sqrt{2+{{x}_{n}}},\ n\ge 1,\ \] and \[\underset{x\to a}{\mathop{\lim }}\,g(x)\] exist done clear
  B)
                  \[\underset{x\to a}{\mathop{\lim }}\,f{{(x)}^{g(x)}}\] exists done clear
  C)
                  \[\underset{x\to a}{\mathop{\lim }}\,\frac{f(x)}{g(x)}\] exists done clear
  D)
                  \[\underset{x\to a}{\mathop{\lim }}\,f(x)g\left( \frac{1}{x} \right)\]exists done clear
  View Solution play_arrow

• question_answer114)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin x+\log (1-x)}{{{x}^{2}}}\] is equal to [Roorkee 1995]
  A)
                  0 done clear
  B)
                  \[\frac{1}{2}\] done clear
  C)
                  \[-\frac{1}{2}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer115)
  If \[a,\ b,\ c,\ d\] are positive, then \[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{1}{a+bx} \right)}^{c+dx}}=\] [EAMCET 1992]
  A)
                  \[{{e}^{d/b}}\] done clear
  B)
                  \[{{e}^{c/a}}\] done clear
  C)
                  \[{{e}^{(c+d)/(a+b)}}\] done clear
  D)
                  \[e\] done clear
  View Solution play_arrow

• question_answer116)
  \[\underset{x\to 0}{\mathop{\lim }}\,{{\left( \frac{1+\tan x}{1+\sin x} \right)}^{\text{cosec }x}}\]is equal to [Kerala (Engg.) 2005] 
  A)
                  \[e\] done clear
  B)
                  \[\frac{1}{e}\] done clear
  C)
                  1 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer117)
  \[\underset{n\to \infty }{\mathop{\lim }}\,{{({{4}^{n}}+{{5}^{n}})}^{1/n}}\]is equal to
  A)
                  \[4\] done clear
  B)
                  5 done clear
  C)
                  \[e\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer118)
  The value of \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{{{x}^{2}}\sin \frac{1}{x}-x}{1-|x|}\] is
  A)
                  \[0\] done clear
  B)
                  1 done clear
  C)
                  ?1 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer119)
  \[\underset{x\to \infty }{\mathop{\lim }}\,\sqrt{\frac{x+\sin x}{x-\cos x}}=\] [Roorkee 1994]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  ?1 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer120)
  \[\underset{x\to 0}{\mathop{\lim }}\,{{\left\{ \tan \left( \frac{\pi }{4}+x \right) \right\}}^{1/x}}=\] [IIT 1993; RPET 2001]
  A)
                  1 done clear
  B)
                  ?1 done clear
  C)
                  \[{{e}^{2}}\] done clear
  D)
                  \[e\] done clear
  View Solution play_arrow

• question_answer121)
  If \[0<x<y\] then \[\underset{n\to \infty }{\mathop{\lim }}\,{{({{y}^{n}}+{{x}^{n}})}^{1/n}}\] is equal to
  A)
                  \[e\] done clear
  B)
                  \[x\] done clear
  C)
                  \[y\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer122)
  The value of \[\underset{x\to \infty }{\mathop{\lim }}\,\sqrt{{{a}^{2}}{{x}^{2}}+ax+1}-\sqrt{{{a}^{2}}{{x}^{2}}+1}\]is
  A)
                  \[\frac{1}{2}\] done clear
  B)
                  1 done clear
  C)
                  \[2\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer123)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{\tan x}}-{{e}^{x}}}{\tan x-x}=\]              [EAMCET 1994; RPET 2001]
  A)
                  1 done clear
  B)
                  \[e\] done clear
  C)
                  \[{{e}^{-1}}\] done clear
  D)
                  \[\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer124)
  If \[f(x)=\sqrt{\frac{x-\sin x}{x+{{\cos }^{2}}x}}\], then \[\underset{x\to \infty }{\mathop{\lim }}\,f(x)\]is [DCE 2000]
  A)
                  0 done clear
  B)
                  \[\infty \] done clear
  C)
                  1 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer125)
  \[\underset{x\to -1}{\mathop{\lim }}\,\frac{\sqrt{\pi }-\sqrt{{{\cos }^{-1}}x}}{\sqrt{x+1}}\]is given by
  A)
                  \[\frac{1}{\sqrt{\pi }}\] done clear
  B)
                  \[\frac{1}{\sqrt{2\pi }}\] done clear
  C)
                  1 done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer126)
  \[\underset{x\to \infty }{\mathop{\lim }}\,\left[ \sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x} \right]\]is equal to
  A)
                  0 done clear
  B)
                  \[\frac{1}{2}\] done clear
  C)
                  \[\frac{1}{p}-\frac{1}{p-1}\] done clear
  D)
                  \[{{e}^{4}}\] done clear
  View Solution play_arrow

• question_answer127)
  If \[f(x)=\frac{2}{x-3},\ g(x)=\frac{x-3}{x+4}\] and \[h(x)=-\frac{2(2x+1)}{{{x}^{2}}+x-12},\] then \[\underset{x\to 3}{\mathop{\lim }}\,[f(x)+g(x)+h(x)]\] is
  A)
                  \[-2\] done clear
  B)
                  \[-1\] done clear
  C)
                  \[-\frac{2}{7}\] done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer128)
  The value of \[\underset{x\to 0}{\mathop{\lim }}\,{{\left( \frac{{{a}^{x}}+{{b}^{x}}+{{c}^{x}}}{3} \right)}^{2/x}}\]; \[(a,\ b,\ c>0)\] is
  A)
                  \[{{(abc)}^{3}}\] done clear
  B)
                  \[abc\] done clear
  C)
                  \[{{(abc)}^{1/3}}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer129)
  The value of  \[\underset{x\to 2}{\mathop{\lim }}\,\frac{\sqrt{1+\sqrt{2+x}}-\sqrt{3}}{x-2}\]is
  A)
                  \[\underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,\frac{\left[ 1-\tan \left( \frac{x}{2} \right) \right]\,[1-\sin x]}{\left[ 1+\tan \left( \frac{x}{2} \right) \right]\,{{[\pi -2x]}^{3}}}\] done clear
  B)
                  \[\frac{1}{4\sqrt{3}}\] done clear
  C)
                  0 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer130)
  The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{1-\cos {{x}^{2}}}}{1-\cos x}\]is
  A)
                  \[\frac{1}{2}\] done clear
  B)
                  \[2\] done clear
  C)
                  \[\sqrt{2}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer131)
  The value of \[\underset{x\to {{0}^{+}}}{\mathop{\lim }}\,{{x}^{m}}{{(\log x)}^{n}},\ m,\ n\in N\]is
  A)
                  0 done clear
  B)
                  \[\frac{m}{n}\] done clear
  C)
                  \[mn\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer132)
  The value of \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{\log x}{{{x}^{n}}},\ n>0\]is
  A)
                  \[0\] done clear
  B)
                  \[1\] done clear
  C)
                  \[\frac{1}{n}\] done clear
  D)
                  \[\frac{1}{n!}\] done clear
  View Solution play_arrow

• question_answer133)
  The value of \[\underset{x\to a}{\mathop{\lim }}\,\frac{\log (x-a)}{\log ({{e}^{x}}-{{e}^{a}})}\]is
  A)
                  1 done clear
  B)
                  ?1 done clear
  C)
                  0 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer134)
  The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{(1+x)}^{1/x}}-e+\frac{1}{2}ex}{{{x}^{2}}}\]is [DCE 2001]
  A)
                  \[\frac{11e}{24}\] done clear
  B)
                  \[\frac{-11e}{24}\] done clear
  C)
                  \[\frac{e}{24}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer135)
  \[\underset{x\to \pi /2}{\mathop{\lim }}\,\left[ x\tan x-\left( \frac{\pi }{2} \right)\sec x \right]=\]
  A)
                  1 done clear
  B)
                  ?1 done clear
  C)
                  0 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer136)
  \[\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{\sin (x+a)+\sin (a-x)-2\sin a}{x\sin x} \right]=\]
  A)
                  \[\sin a\] done clear
  B)
                  \[\cos a\] done clear
  C)
                  \[-\sin a\] done clear
  D)
                  \[\frac{1}{2}\cos a\] done clear
  View Solution play_arrow

• question_answer137)
  \[\underset{x\to 0}{\mathop{\lim }}\,{{\left( \frac{1+5{{x}^{2}}}{1+3{{x}^{2}}} \right)}^{1/{{x}^{2}}}}=\] [IIT 1996; DCE 2001]
  A)
                  \[{{e}^{2}}\] done clear
  B)
                  \[e\] done clear
  C)
                  \[{{e}^{-2}}\] done clear
  D)
                  \[{{e}^{-1}}\] done clear
  View Solution play_arrow

• question_answer138)
  \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{(2x-3)(3x-4)}{(4x-5)(5x-6)}=\] [MP PET 1996]
  A)
                  0 done clear
  B)
                  1/10 done clear
  C)
                  1/5 done clear
  D)
                  3/10 done clear
  View Solution play_arrow

• question_answer139)
  If \[f(x)=\frac{\sin ({{e}^{x-2}}-1)}{\log (x-1)},\]then \[\underset{x\to 2}{\mathop{\lim }}\,f(x)\]is given by
  A)
                  ?2 done clear
  B)
                  ?1 done clear
  C)
                  0 done clear
  D)
                  1 done clear
  View Solution play_arrow

• question_answer140)
  \[\underset{x\to \infty }{\mathop{\lim }}\,(\sqrt{{{x}^{2}}+8x+3}-\sqrt{{{x}^{2}}+4x+3})=\]        [MP PET 1997]
  A)
                  0 done clear
  B)
                  \[\infty \] done clear
  C)
                  2 done clear
  D)
                  \[\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer141)
  If \[\underset{x\to 5}{\mathop{\lim }}\,\frac{{{x}^{k}}-{{5}^{k}}}{x-5}=500\], then the positve integral value of k is [MP PET 1998]
  A)
                  3 done clear
  B)
                  4 done clear
  C)
                  5 done clear
  D)
                  6 done clear
  View Solution play_arrow

• question_answer142)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sqrt{1-{{x}^{2}}}-\sqrt{1+{{x}^{2}}}}{{{x}^{2}}}\] is equal to [MP PET 1999]
  A)
                  1 done clear
  B)
                  ?1 done clear
  C)
                  ?2 done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer143)
  If \[f(x)=\left\{ \begin{align}   & x,\ \ \text{if }x\text{ is rational } \\  & -x,\ \text{if }x\text{ is irrational} \\ \end{align} \right.,\] then \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\]is                 [Kurukshetra CEE 1998; UPSEAT 2004]
  A)
                  Equal to 0 done clear
  B)
                  Equal to 1 done clear
  C)
                  Equal to ?1 done clear
  D)
                  Indeterminate done clear
  View Solution play_arrow

• question_answer144)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{x{{e}^{x}}-\log (1+x)}{{{x}^{2}}}\] equals [RPET 1996]
  A)
                  \[\frac{2}{3}\] done clear
  B)
                  \[\frac{1}{3}\] done clear
  C)
                  \[\frac{1}{2}\] done clear
  D)
                  \[\frac{3}{2}\] done clear
  View Solution play_arrow

• question_answer145)
  The value of \[\underset{x\to -\infty }{\mathop{\lim }}\,\frac{\sqrt{4{{x}^{2}}+5x+8}}{4x+5}\]is [Roorkee 1998]
  A)
                  \[-1/2\] done clear
  B)
                  0 done clear
  C)
                  \[1/2\] done clear
  D)
                  1 done clear
  View Solution play_arrow

• question_answer146)
  \[\underset{x\to \infty }{\mathop{\lim }}\,{{\left[ 1+\frac{1}{mx} \right]}^{x}}\]equal to              [Kurukshetra CEE 1998]
  A)
                  \[{{e}^{1/m}}\] done clear
  B)
                  \[{{e}^{-1/m}}\] done clear
  C)
                  \[{{e}^{m}}\] done clear
  D)
                  \[{{m}^{e}}\] done clear
  View Solution play_arrow

• question_answer147)
  Let the function f be defined by the equation \[f(x)=\left\{ \begin{align}   & 3x\ \ \ \ \ \ \text{if}\ 0\le x\le 1 \\  & 5-3x\ \ \text{if}\ \text{1}<x\le 2 \\ \end{align} \right.,\]then [SCRA 1996]
  A)
                  \[\underset{x\to 1}{\mathop{\lim }}\,f(x)=f(1)\] done clear
  B)
                  \[\underset{x\to 1}{\mathop{\lim }}\,f(x)=3\] done clear
  C)
                  \[\underset{x\to 1}{\mathop{\lim }}\,f(x)=2\] done clear
  D)
                  \[\underset{x\to 1}{\mathop{\lim }}\,f(x)\]does not exist done clear
  View Solution play_arrow

• question_answer148)
  The value of the limit of \[\frac{{{x}^{3}}-8}{{{x}^{2}}-4}\]as x tends to 2 is [SCRA 1996]
  A)
                  3 done clear
  B)
                  \[\frac{3}{2}\] done clear
  C)
                  1 done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer149)
  The value of the limit of \[\frac{{{x}^{3}}-{{x}^{2}}-18}{x-3}\]as x tends to 3 is [SCRA 1996]
  A)
                  3 done clear
  B)
                  9 done clear
  C)
                  18 done clear
  D)
                  21 done clear
  View Solution play_arrow

• question_answer150)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\tan }^{-1}}x}{x}\]is [SCRA 1996]
  A)
                  0 done clear
  B)
                  \[\infty \] done clear
  C)
                  ?1 done clear
  D)
                  1 done clear
  View Solution play_arrow

• question_answer151)
  \[x=1\] is equal to [SCRA 1996]
  A)
                  \[\frac{2}{3}\] done clear
  B)
                  \[1\] done clear
  C)
                  0 done clear
  D)
                  \[\infty \] done clear
  View Solution play_arrow

• question_answer152)
  \[\underset{x\to 0}{\mathop{\lim }}\,\sin \left( \frac{1}{x} \right)\] is [SCRA 1996]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  ?1 done clear
  D)
                  Does not exist done clear
  View Solution play_arrow

• question_answer153)
  \[\underset{x\to 4}{\mathop{\lim }}\,\left[ \frac{{{x}^{3/2}}-8}{x-4} \right]=\] [DCE 1999]
  A)
                  3/2 done clear
  B)
                  3 done clear
  C)
                  2/3 done clear
  D)
                  1/3 done clear
  View Solution play_arrow

• question_answer154)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{e}^{\frac{1}{x}}}}{{{e}^{\left( \frac{1}{x}+1 \right)}}}=\] [DCE 1999]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  Does not exist done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer155)
  The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{x\cos x-\log (1+x)}{{{x}^{2}}}\]is [RPET 1999]
  A)
                  ½ done clear
  B)
                  0 done clear
  C)
                  1 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer156)
  The value of \[\underset{a\to 0}{\mathop{\lim }}\,\frac{\sin a-\tan a}{{{\sin }^{3}}a}\]will be [UPSEAT 1999]
  A)
                  \[-\frac{1}{2}\] done clear
  B)
                  \[\frac{1}{2}\] done clear
  C)
                  1 done clear
  D)
                  ?1 done clear
  View Solution play_arrow

• question_answer157)
  \[\underset{n\to \infty }{\mathop{\lim }}\,{{\left( \frac{n}{n+y} \right)}^{n}}\] equals [AMU 1999]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  1/v done clear
  D)
                  \[{{e}^{-y}}\] done clear
  View Solution play_arrow

• question_answer158)
  If \[f(x)=\left\{ \begin{align}   & x\ :\ x<0 \\  & 1\ :\ x=0 \\  & {{x}^{2}}\ :\ x>0 \\ \end{align} \right.,\]then \[\underset{x\to 0}{\mathop{\lim }}\,f(x)=\] [DCE 2000]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  2 done clear
  D)
                  Does not exist done clear
  View Solution play_arrow

• question_answer159)
  If \[f(x)=\left\{ \begin{align}   & \sin x,x\ne n\pi ,n\in Z \\  & \,\,\,\,\,\,0,\,\,\text{otherwise} \\ \end{align} \right.\] and \[g(x)=\left\{ \begin{align}   & {{x}^{2}}+1,x\ne 0,\,2 \\  & \,\,\,\,\,\,\,\,4,x=0 \\  & \,\,\,\,\,\,\,\,\,5,x=2 \\ \end{align} \right.\] then \[\underset{x\to 0}{\mathop{\lim }}\,g\{f(x)\}=\] [Karnataka CET 2000]
  A)
                  1 done clear
  B)
                  0 done clear
  C)
                  \[\frac{1}{2}\] done clear
  D)
                  \[\frac{1}{4}\] done clear
  View Solution play_arrow

• question_answer160)
  \[\underset{x\to 1}{\mathop{\lim }}\,\frac{1+\log x-x}{1-2x+{{x}^{2}}}=\]  [Karnataka CET 2000; Pb. CET 2001]
  A)
                  1 done clear
  B)
                  ?1 done clear
  C)
                  0 done clear
  D)
                  \[-\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer161)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{a}^{\sin x}}-1}{{{b}^{\sin x}}-1}=\] [Karnataka CET 2000]
  A)
                  \[\frac{a}{b}\] done clear
  B)
                  \[\frac{b}{a}\] done clear
  C)
                  \[\frac{\log a}{\log b}\] done clear
  D)
                  \[\frac{\log b}{\log a}\] done clear
  View Solution play_arrow

• question_answer162)
  The value of \[\underset{x\to 2}{\mathop{\lim }}\,\frac{{{3}^{x/2}}-3}{{{3}^{x}}-9}\] is [MP PET 2000]
  A)
                  0 done clear
  B)
                  1/3 done clear
  C)
                  \[1/6\] done clear
  D)
                  \[\ln 3\] done clear
  View Solution play_arrow

• question_answer163)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{\sin }^{-1}}x-{{\tan }^{-1}}x}{{{x}^{3}}}\] is equal to [RPET 2000]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  ?1 done clear
  D)
                  \[1/2\] done clear
  View Solution play_arrow

• question_answer164)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{x\tan 2x-2x\tan x}{{{(1-\cos 2x)}^{2}}}\] is [IIT 1999]
  A)
                  2 done clear
  B)
                  ?2 done clear
  C)
                  \[\frac{1}{2}\] done clear
  D)
                  \[-\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer165)
  The value of \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{(1-\cos 2x)\sin 5x}{{{x}^{2}}\sin 3x}\] is [MP PET 2000; UPSEAT 2000; Karnataka CET 2002]
  A)
                  10/3 done clear
  B)
                  3/10 done clear
  C)
                  6/5 done clear
  D)
                  5/6                done clear
  View Solution play_arrow

• question_answer166)
  \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{\text{ln}\,(\cos x)}{{{x}^{2}}}\] is equal to [AMU 2000]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  \[\frac{1}{2}\] done clear
  D)
                  \[-\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer167)
  For \[x\in R,\,\,\,\underset{x\to \infty }{\mathop{\lim }}\,\,{{\left( \frac{x-3}{x+2} \right)}^{x}}\] is equal to [IIT Screening 2000]
  A)
                  e done clear
  B)
                  \[{{e}^{-1}}\] done clear
  C)
                  \[{{e}^{-5}}\] done clear
  D)
                  \[{{e}^{5}}\] done clear
  View Solution play_arrow

• question_answer168)
  The value of \[\underset{x\to 0}{\mathop{\lim }}\,\,\left( \frac{{{e}^{x}}-1}{x} \right)\] is               [Karnataka CET 2001]
  A)
                  1/2 done clear
  B)
                  \[\infty \] done clear
  C)
                  1 done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer169)
  The value of \[\underset{x\to 0}{\mathop{\lim }}\,\,\left[ \frac{\sqrt{a+x}-\sqrt{a-x}}{x} \right]\] is  [Karnataka CET 2001]
  A)
                  1 done clear
  B)
                  0 done clear
  C)
                  \[\sqrt{a}\] done clear
  D)
                  \[1/\sqrt{a}\] done clear
  View Solution play_arrow

• question_answer170)
  \[\underset{\alpha \to \beta }{\mathop{\lim }}\,\left[ \frac{{{\sin }^{2}}\alpha -{{\sin }^{2}}\beta }{{{\alpha }^{2}}-{{\beta }^{2}}} \right]=\] [MP PET 2001]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  \[\frac{\sin \beta }{\beta }\] done clear
  D)
                  \[\frac{\sin 2\beta }{2\beta }\] done clear
  View Solution play_arrow

• question_answer171)
  \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{{{(1+x)}^{1/x}}-e}{x}\] equals [UPSEAT 2001]
  A)
                  \[\pi /2\] done clear
  B)
                  0 done clear
  C)
                  \[2/e\] done clear
  D)
                  ?\[e/2\] done clear
  View Solution play_arrow

• question_answer172)
  \[\underset{x\to 1}{\mathop{\lim }}\,\frac{1+\cos \pi \,x}{{{\tan }^{2}}\pi \,x}\] is equal to [AMU 2001]
  A)
                  0 done clear
  B)
                  1/2 done clear
  C)
                  1 done clear
  D)
                  2 done clear
  View Solution play_arrow

• question_answer173)
  \[\underset{m\to \infty }{\mathop{\lim }}\,\,{{\left( \cos \frac{x}{m} \right)}^{m}}=\]    [AMU 2001]
  A)
                  0 done clear
  B)
                  e done clear
  C)
                  1/e done clear
  D)
                  1 done clear
  View Solution play_arrow

• question_answer174)
  \[\underset{x\to \infty }{\mathop{\lim }}\,\,{{\left( \frac{x+a}{x+b} \right)}^{x+b}}=\] [EAMCET 2001]
  A)
                  1 done clear
  B)
                  \[{{e}^{b-a}}\] done clear
  C)
                  \[{{e}^{a-b}}\] done clear
  D)
                  \[{{e}^{b}}\] done clear
  View Solution play_arrow

• question_answer175)
  \[\underset{x\to \pi /2}{\mathop{\lim }}\,\frac{{{a}^{\cot x}}-{{a}^{\cos x}}}{\cot x-\cos x}=\]                        [Kerala (Engg.) 2001; J & K 2005]
  A)
                  \[\log a\] done clear
  B)
                  \[\log 2\] done clear
  C)
                  a done clear
  D)
                  log x done clear
  View Solution play_arrow

• question_answer176)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin (\pi {{\cos }^{2}}x)}{{{x}^{2}}}=\] [IIT Screening 2001;UPSEAT 2001; MP PET 2002]
  A)
                  \[(-1,1)\] done clear
  B)
                  \[\pi \] done clear
  C)
                  \[\pi /2\] done clear
  D)
                  1 done clear
  View Solution play_arrow

• question_answer177)
  \[\underset{x\to 3}{\mathop{\lim }}\,\,[x]=\], (where [.] = greatest integer function) [DCE 2002]
  A)
                  2 done clear
  B)
                  3 done clear
  C)
                  Does not exist done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer178)
  If \[f(x)\,=\left| \,\begin{matrix}    \sin x & \cos x & \tan x  \\    {{x}^{3}} & {{x}^{2}} & x  \\    2x & 1 & 1  \\ \end{matrix}\, \right|\], then \[\underset{x\to 0}{\mathop{\lim }}\,\frac{f(x)}{{{x}^{2}}}\] is [Karnataka CET 2002]
  A)
                  3 done clear
  B)
                  ?1 done clear
  C)
                  0 done clear
  D)
                  1 done clear
  View Solution play_arrow

• question_answer179)
  \[x=1\] [MP PET 2002]
  A)
                  \[{{\log }_{e}}3\] done clear
  B)
                  0 done clear
  C)
                  1 done clear
  D)
                  \[f(x)\] done clear
  View Solution play_arrow

• question_answer180)
  \[\underset{x\to 0}{\mathop{\lim }}\,\,\,\cos \frac{1}{x}\] [UPSEAT 2002]
  A)
                  Is continuous at \[x=0\] done clear
  B)
                  Is differentiable at \[(3,\,\,1)\] done clear
  C)
                  Does not exist done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer181)
  Let \[f(x)=4\] and \[f'(x)=4\], then \[\underset{x\to 2}{\mathop{\lim }}\,\,\frac{xf(2)-2f(x)}{x-2}\] equals [AIEEE 2002]
  A)
                  2 done clear
  B)
                  ? 2 done clear
  C)
                  ? 4 done clear
  D)
                  3 done clear
  View Solution play_arrow

• question_answer182)
  \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{\log {{x}^{n}}-[x]}{[x]},\,n\in N,\,\]\[\,(\,[x]\] denotes greatest integer less than or equal to x) [AIEEE 2002]
  A)
                  Has value  ?1 done clear
  B)
                  Has value 0 done clear
  C)
                  Has value 1 done clear
  D)
                  Does not exist done clear
  View Solution play_arrow

• question_answer183)
  If \[f(1)\,=1,\,{f}'\,(1)\,=2\], then \[\underset{x\to 1}{\mathop{\lim }}\,\frac{\sqrt{f(x)}-1}{\sqrt{x}-1}\] is [AIEEE 2002]
  A)
                  2 done clear
  B)
                  4 done clear
  C)
                  1 done clear
  D)
                  ½ done clear
  View Solution play_arrow

• question_answer184)
  \[\underset{n\to \infty }{\mathop{\lim }}\,\,{{\left( \frac{{{n}^{2}}-n+1}{{{n}^{2}}-n-1} \right)}^{n(n-1)}}=\] [AMU 2002]
  A)
                  e done clear
  B)
                  \[{{e}^{2}}\] done clear
  C)
                  \[{{e}^{-1}}\] done clear
  D)
                  1 done clear
  View Solution play_arrow

• question_answer185)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{4}^{x}}-{{9}^{x}}}{x({{4}^{x}}+{{9}^{x}})}=\] [EAMCET 2002]
  A)
                  \[\log \left( \frac{2}{3} \right)\]            done clear
  B)
                  \[\frac{1}{2}\log \left( \frac{3}{2} \right)\] done clear
  C)
                  \[\frac{1}{2}\log \left( \frac{2}{3} \right)\] done clear
  D)
                  \[\log \,\left( \frac{3}{2} \right)\] done clear
  View Solution play_arrow

• question_answer186)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{a}^{x}}-{{b}^{x}}}{{{e}^{x}}-1}\]= [Kerala (Engg.) 2002]
  A)
                  \[\log \left( \frac{a}{b} \right)\] done clear
  B)
                  \[\log \left( \frac{b}{a} \right)\] done clear
  C)
                  \[\log (a\,b)\] done clear
  D)
                  \[\log \,(a+\,b)\] done clear
  View Solution play_arrow

• question_answer187)
  If \[f(x)\,={{\cot }^{-1}}\left( \frac{3x-{{x}^{3}}}{1-3{{x}^{2}}} \right)\]  and x\[g(x)={{\cos }^{-1}}\left( \frac{1-{{x}^{2}}}{1+{{x}^{2}}} \right)\], then \[\underset{x\to a}{\mathop{\lim }}\,\frac{f(x)-f(a)}{g(x)\,-g(a)},\] \[0<\,a<\frac{1}{2}\] is [Orissa JEE 2002]
  A)
                  \[\frac{3}{2(1+{{a}^{2}})}\] done clear
  B)
                  \[\frac{3}{2(1+{{x}^{2}})}\] done clear
  C)
                  \[\frac{3}{2}\] done clear
  D)
                  \[-\frac{3}{2}\] done clear
  View Solution play_arrow

• question_answer188)
  \[\underset{x\to -2}{\mathop{\lim }}\,\frac{{{\sin }^{-1}}(x+2)}{{{x}^{2}}+2x}\] is equal to [Orissa JEE 2002]
  A)
                  0 done clear
  B)
                  \[\infty \] done clear
  C)
                  ?1/2 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer189)
  \[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( \frac{x+3}{x+1} \right)}^{x+1}}=\]        [RPET 2003, UPSEAT 2003]
  A)
                  \[{{e}^{2}}\] done clear
  B)
                  \[{{e}^{3}}\] done clear
  C)
                  e done clear
  D)
                  \[{{e}^{-1}}\] done clear
  View Solution play_arrow

• question_answer190)
  \[\underset{x\to 0}{\mathop{\lim }}\,{{(1-ax)}^{\frac{1}{x}}}=\] [Karnataka CET 2003]
  A)
                  e done clear
  B)
                  \[{{e}^{-a}}\] done clear
  C)
                  1 done clear
  D)
                  \[{{e}^{a}}\] done clear
  View Solution play_arrow

• question_answer191)
  The value of \[\underset{x\to 7}{\mathop{\lim }}\,\frac{2-\sqrt{x-3}}{{{x}^{2}}-49}\] is [MP PET 2003]
  A)
                  \[\frac{2}{9}\] done clear
  B)
                  \[-\frac{2}{49}\] done clear
  C)
                  \[\frac{1}{56}\] done clear
  D)
                  \[-\frac{1}{56}\] done clear
  View Solution play_arrow

• question_answer192)
  If \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\log (3+x)\,-\log (3-x)}{x}=k,\,\] then the value of k is [AIEEE 2003]
  A)
                  0 done clear
  B)
                  \[-\frac{1}{3}\] done clear
  C)
                  \[\frac{2}{3}\] done clear
  D)
                  \[-\frac{2}{3}\] done clear
  View Solution play_arrow

• question_answer193)
  If \[\underset{x\to 0}{\mathop{\lim }}\,\frac{[(a-n)\,nx-\tan x]\sin nx}{{{x}^{2}}}=0,\] where n is non zero real number, then a is equal to [IIT Screening 2003]
  A)
                  0 done clear
  B)
                  \[\frac{n+1}{n}\] done clear
  C)
                  n done clear
  D)
                  \[n+\frac{1}{n}\] done clear
  View Solution play_arrow

• question_answer194)
  Given that\[f'\](2)=6 and \[{f}'(1)=4)=\], then \[\underset{h\to 0}{\mathop{\lim }}\,\frac{f(2h+2+{{h}^{2}})-f(2)}{f(h-{{h}^{2}}+1)-f(1)}=\] [IIT Screening 2003]
  A)
                  Does not exist done clear
  B)
                  Is equal to ? 3/2 done clear
  C)
                  Is equal to 3/2 done clear
  D)
                  Is equal to 3 done clear
  View Solution play_arrow

• question_answer195)
  \[\underset{x\to 0}{\mathop{\lim }}\,\,\frac{{{e}^{x}}-{{e}^{-x}}}{\sin x}\] is [Kurukshetra CEE 2002]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  2 done clear
  D)
                  Non existent done clear
  View Solution play_arrow

• question_answer196)
  \[\underset{x\to \pi /6}{\mathop{\lim }}\,\left[ \frac{3\sin x-\sqrt{3}\cos x}{6x-\pi } \right]=\] [EAMCET 2003]
  A)
                  \[\sqrt{3}\] done clear
  B)
                  \[1/\sqrt{3}\] done clear
  C)
                  \[-\sqrt{3}\] done clear
  D)
                  \[-1/\sqrt{3}\] done clear
  View Solution play_arrow

• question_answer197)
  . \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\cos (\sin x)-1}{{{x}^{2}}}=\] [Orissa JEE 2003]
  A)
                  1 done clear
  B)
                  ? 1 done clear
  C)
                  ½ done clear
  D)
                  ?1/2 done clear
  View Solution play_arrow

• question_answer198)
  \[\underset{n\to \infty }{\mathop{\lim }}\,{{({{3}^{n}}+{{4}^{n}})}^{\frac{1}{n}}}=\] [Karnataka CET 2003]
  A)
                  3 done clear
  B)
                  4 done clear
  C)
                  \[\infty \] done clear
  D)
                  e done clear
  View Solution play_arrow

• question_answer199)
  If \[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( 1+\frac{a}{x}+\frac{b}{{{x}^{2}}} \right)}^{2x}}={{e}^{2}},\]then the values of a and b are [AIEEE 2004]
  A)
                  \[a=1,\ b=2\] done clear
  B)
                  \[\cos (|x|)\,-|x|\] done clear
  C)
                  \[a\in R,\ b=2\] done clear
  D)
                  \[a\in R,\ b\in R\] done clear
  View Solution play_arrow

• question_answer200)
  \[\underset{\theta \to \frac{\pi }{2}}{\mathop{\lim }}\,\frac{\frac{\pi }{2}-\theta }{\cot \theta }\]= [Karnataka CET 2004]
  A)
                  0 done clear
  B)
                  ?1 done clear
  C)
                  1 done clear
  D)
                  \[\infty \] done clear
  View Solution play_arrow

• question_answer201)
  \[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( 1-\frac{4}{x-1} \right)}^{3x-1}}=\] [Karnataka CET 2004]
  A)
                  \[{{e}^{12}}\] done clear
  B)
                  \[{{e}^{-12}}\] done clear
  C)
                  \[{{e}^{4}}\] done clear
  D)
                  \[{{e}^{3}}\] done clear
  View Solution play_arrow

• question_answer202)
  \[\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{{{e}^{x}}-{{e}^{\sin x}}}{x-\sin x} \right]\]is equal to [UPSEAT 2004]
  A)
                  ?1 done clear
  B)
                  0 done clear
  C)
                  1 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer203)
  The value of \[\underset{x\to -1}{\mathop{\lim }}\,\frac{{{x}^{2}}+3x+2}{{{x}^{2}}+4x+3}\]is equal to [Pb. CET 2000]
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  2 done clear
  D)
                  ½ done clear
  View Solution play_arrow

• question_answer204)
  The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{2}{x}\log (1+x)\] is equal to [Pb. CET 2000]
  A)
                  e done clear
  B)
                  \[{{e}^{2}}\] done clear
  C)
                  \[\frac{1}{2}\] done clear
  D)
                  2 done clear
  View Solution play_arrow

• question_answer205)
  The value of \[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( \frac{3x-4}{3x+2} \right)}^{\frac{x+1}{3}}}\] is equal to [Pb. CET 2004]
  A)
                  \[{{e}^{-1/3}}\] done clear
  B)
                  \[{{e}^{-2/3}}\] done clear
  C)
                  \[{{e}^{-1}}\] done clear
  D)
                  \[{{e}^{-2}}\] done clear
  View Solution play_arrow

• question_answer206)
  The value  of \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{(x+1)(3x+4)}{{{x}^{2}}(x-8)}\] is equal to [Pb. CET 2002]
  A)
                  2 done clear
  B)
                  3 done clear
  C)
                  1 done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer207)
  If \[f(x)=\left\{ \begin{align}   & \frac{\sin [x]}{[x]},\text{ when }[x]\ne 0 \\  & \,\,\,\,\,\,\,\,\,0,\text{ when }[x]=0 \\ \end{align} \right.\] where [x] is greatest integer function, then \[\underset{x\to 0}{\mathop{\lim }}\,f(x)=\] [IIT 1985; RPET 1995]
  A)
                  ?1 done clear
  B)
                  1 done clear
  C)
                  0 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer208)
  If \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1-{{(10)}^{n}}}{1+{{(10)}^{n+1}}}=\frac{-\alpha }{10}\], then give the value of \[\alpha \] is [Orissa JEE 2005]
  A)
                  0 done clear
  B)
                  ?1 done clear
  C)
                  1 done clear
  D)
                  2 done clear
  View Solution play_arrow

• question_answer209)
  The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\log [1+{{x}^{3}}]}{{{\sin }^{3}}x}=\] [AMU 2005]
  A)
                  0 done clear
  B)
              1 done clear
  C)
                  3 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer210)
  \[\underset{\theta \to 0}{\mathop{\lim }}\,\frac{4\theta (\tan \theta -2\theta \tan \theta )}{{{(1-\cos 2\theta )}^{2}}}\]is [Orissa JEE 2005]
  A)
                  \[1/\sqrt{2}\] done clear
  B)
                  1/2 done clear
  C)
                  1 done clear
  D)
                  2 done clear
  View Solution play_arrow

• question_answer211)
  The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{27}^{x}}-{{9}^{x}}-{{3}^{x}}+1}{\sqrt{5}-\sqrt{4+\cos x}}\] is [J & K 2005]
  A)
                  \[\sqrt{5}{{(\log 3)}^{2}}\] done clear
  B)
                  \[8\sqrt{5}\log 3\] done clear
  C)
                  \[16\sqrt{5}\log 3\] done clear
  D)
                  \[8\sqrt{5}{{(\log 3)}^{2}}\] done clear
  View Solution play_arrow

• question_answer212)
  The value of \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{x}^{n}}}{{{x}^{n}}+1}\] where \[x<-1\] is [J & K 2005]
  A)
                  ½ done clear
  B)
                  ?1/2 done clear
  C)
                  1 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer213)
  The value of \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+...+\frac{1}{(2n-1)(2n+1)}\] is equal to [DCE 2005]
  A)
                  ½ done clear
  B)
                  1/3 done clear
  C)
                  ¼ done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer214)
  The value of the constant \[\alpha \] and \[\beta \] such that \[\underset{x\to \infty }{\mathop{\lim }}\,\left( \frac{{{x}^{2}}+1}{x+1}-\alpha x-\beta  \right)=0\] are respectively [Orissa JEE 2005]
  A)
                  (1, 1) done clear
  B)
                  (?1, 1) done clear
  C)
                  (1, ?1) done clear
  D)
                  (0, 1) done clear
  View Solution play_arrow

• question_answer215)
  Let \[f:R\to R\]be a differentiable function having \[f(2)=6,f'(2)=\left( \frac{1}{48} \right).\] Then \[\underset{x\to 2}{\mathop{\lim }}\,\int\limits_{6}^{f(x)}{\frac{4{{t}^{3}}}{x-2}}\]dt equals [AIEEE 2005]
  A)
                  12 done clear
  B)
                  18 done clear
  C)
                  24 done clear
  D)
                  36 done clear
  View Solution play_arrow

• question_answer216)
  \[[.]\]is equal to               [IIT 1984; DCE 2000; Pb. CET 2000]
  A)
                  0 done clear
  B)
                  \[-\frac{1}{2}\] done clear
  C)
                  \[\log \left( \frac{2}{3} \right)\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer217)
  \[\underset{n\to \infty }{\mathop{\lim }}\,\left[ \frac{1}{{{n}^{3}}+1}+\frac{4}{{{n}^{3}}+1}+\frac{9}{{{n}^{3}}+1}+........+\frac{{{n}^{2}}}{{{n}^{3}}+1} \right]=\]
  A)
                  \[1\] done clear
  B)
                  2/3 done clear
  C)
                  1/3 done clear
  D)
                  \[0\] done clear
  View Solution play_arrow

• question_answer218)
  If \[{{S}_{n}}=\sum\limits_{k=1}^{n}{{{a}_{k}}}\]and\[\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}=a,\]then \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{S}_{n+1}}-{{S}_{n}}}{\sqrt{\sum\limits_{k=1}^{n}{k}}}\]is equal to
  A)
                  0 done clear
  B)
                  a done clear
  C)
                  \[\sqrt{2}a\] done clear
  D)
                  \[2a\] done clear
  View Solution play_arrow

• question_answer219)
  If \[{{a}_{1}}=1\] and \[{{a}_{n+1}}=\frac{4+3{{a}_{n}}}{3+2{{a}_{n}}},\ n\ge 1\] and if \[-\frac{1}{3}\], then the value of a is
  A)
                  \[\sqrt{2}\] done clear
  B)
                  \[-\sqrt{2}\] done clear
  C)
                  2 done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer220)
  The value of \[\underset{n\to \infty }{\mathop{\lim }}\,\cos \left( \frac{x}{2} \right)\cos \left( \frac{x}{4} \right)\cos \left( \frac{x}{8} \right)...\cos \left( \frac{x}{{{2}^{n}}} \right)\] is
  A)
                  1 done clear
  B)
                  \[\frac{\sin x}{x}\] done clear
  C)
                  \[\frac{x}{\sin x}\] done clear
  D)
                  None of these done clear
  View Solution play_arrow

• question_answer221)
  \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1}{2}+\frac{1}{{{2}^{2}}}+\frac{1}{{{2}^{3}}}+...+\frac{1}{{{2}^{n}}}\]equals [RPET 1996]
  A)
                  2 done clear
  B)
                  ?1 done clear
  C)
                  1 done clear
  D)
                  3 done clear
  View Solution play_arrow

• question_answer222)
  \[\underset{n\to \infty }{\mathop{\lim }}\,\left\{ \frac{1}{{{n}^{2}}}+\frac{2}{{{n}^{2}}}+\frac{3}{{{n}^{2}}}+......+\frac{n}{{{n}^{2}}} \right\}\]is [SCRA 1996]
  A)
                  ½ done clear
  B)
                  0 done clear
  C)
                  1 done clear
  D)
                  \[\infty \] done clear
  View Solution play_arrow

• question_answer223)
  The value of  \[\underset{n\,\to \,\infty }{\mathop{\lim }}\,\frac{1-{{n}^{2}}}{\sum n}\] will be [UPSEAT 1999]
  A)
                  ? 2 done clear
  B)
                  ? 1 done clear
  C)
                  2 done clear
  D)
                  1 done clear
  View Solution play_arrow

• question_answer224)
  If \[{{x}_{n}}=\frac{1-2+3-4+5-6+.....-2n}{\sqrt{{{n}^{2}}+1}+\sqrt{4{{n}^{2}}-1}},\] then \[\underset{n\to \infty }{\mathop{\lim }}\,{{x}_{n}}\] is equal to [AMU 2000]
  A)
                  \[\frac{1}{3}\] done clear
  B)
                  \[-\frac{2}{3}\] done clear
  C)
                  \[\frac{2}{3}\] done clear
  D)
                  1 done clear
  View Solution play_arrow

• question_answer225)
  \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{{{(x+1)}^{10}}+{{(x+2)}^{10}}+.....+{{(x+100)}^{10}}}{{{x}^{10}}+{{10}^{10}}}\] is equal to
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  10 done clear
  D)
                  100 done clear
  View Solution play_arrow

• question_answer226)
  The value of \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1+2+3+....n}{{{n}^{2}}+100}\]is equal         [Pb. CET 2002]
  A)
                  \[\infty \] done clear
  B)
                  \[\frac{1}{2}\] done clear
  C)
                  2 done clear
  D)
                  0 done clear
  View Solution play_arrow

• question_answer227)
  The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\int_{0}^{x}{\cos {{t}^{2}}}}{x}\,dt\] is
  A)
                  0 done clear
  B)
                  1 done clear
  C)
                  \[-1\] done clear
  D)
                  None of these done clear
  View Solution play_arrow