JEE Main & Advanced Mathematics Differentiation Question Bank

done Self Evaluation Test - Limits and Derivatives Total Questions - 70

• question_answer1)
  If \[\underset{x\to 0}{\mathop{\lim }}\,\,\,kx\cos ec\,x=\underset{x\to 0}{\mathop{\lim }}\,x\cos ec\,\,kx,\] then \[k=\]
  A)
  1 done clear
  B)
  -1 done clear
  C)
  \[\pm 1\] done clear
  D)
  \[\pm 2\] done clear
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• question_answer2)
  Let \[f(x)=\alpha (x)\beta (x)\gamma (x)\] for all real x, where \[\alpha (x),\beta (x)\] and \[\gamma (x)\] are differentiable functions of\[x.\] If \[f'(2)=18f(2),\alpha '(2)=3\alpha (2),\beta '(2)=-4\beta (2)\] and \[\gamma '(2)-k\gamma (2),\] then the value of k is
  A)
  14 done clear
  B)
  16 done clear
  C)
  19 done clear
  D)
  None of these   done clear
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• question_answer3)
  \[\underset{n\,\to \,\infty }{\mathop{\lim }}\,\frac{{{5}^{n+1}}+{{3}^{n}}-{{2}^{2n}}}{{{5}^{n}}+{{2}^{n}}+{{3}^{2n+3}}}\] is equal to
  A)
  5 done clear
  B)
  3 done clear
  C)
  1 done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer4)
  If \[\{x\}\]denotes the fractional part of x, then \[\underset{x\to [a]}{\mathop{\lim }}\,\frac{{{e}^{\{x\}}}-\{x\}-1}{{{\{x\}}^{2}}},\] Where [a] denotes the integral part of a, is equal to
  A)
  0 done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  \[e-2\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer5)
  \[\underset{x\to 0}{\mathop{\lim }}\,\left[ \min ({{y}^{2}}-4y+11)\frac{\sin x}{x} \right]\] (where [.] denotes the greatest integer function) is
  A)
  5 done clear
  B)
  6 done clear
  C)
  7 done clear
  D)
  Does not exist   done clear
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• question_answer6)
  \[\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{\sin (sgn(x))}{(sgn(x))} \right],\] where [.] denotes the greatest integer function, is equal to
  A)
  0 done clear
  B)
  1 done clear
  C)
  -1 done clear
  D)
  Does not exist done clear
  View Solution play_arrow

• question_answer7)
  For the function \[f(x)=\frac{{{x}^{100}}}{100}+\frac{{{x}^{99}}}{99}+...\frac{{{x}^{2}}}{2}+x+1.\] \[f'(1)=mf'(0),\] Where m is equal to
  A)
  50 done clear
  B)
  0 done clear
  C)
  100 done clear
  D)
  200 done clear
  View Solution play_arrow

• question_answer8)
  If \[y=(1+{{x}^{1/4}})(1+{{x}^{1/2}})(1-{{x}^{1/4}}),\] then \[\frac{dy}{dx}\] is equal to
  A)
  1 done clear
  B)
  -1 done clear
  C)
  x done clear
  D)
  \[\sqrt{x}\] done clear
  View Solution play_arrow

• question_answer9)
  If \[f(x)=\left\{ \begin{matrix}    {{x}^{n}}\sin (1/{{x}^{2}}),x\ne 0  \\    0,x=0  \\ \end{matrix} \right.\], \[(n\in I)\], then
  A)
  \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\] exists for \[n>1\] done clear
  B)
  \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\] exists for \[n<0\] done clear
  C)
  \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\] Does not exist for any value of n done clear
  D)
  \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\] cannot be determined done clear
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• question_answer10)
  If \[x>0\] and \[g\] is  a bounded function, then \[\underset{n\,\to \,\infty }{\mathop{\lim }}\,\frac{f(x){{e}^{nx}}+g(x)}{{{e}^{nx}}+1}\] is
  A)
  0 done clear
  B)
  \[f(x)\] done clear
  C)
  \[g(x)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer11)
  Let the sequence \[<{{b}_{n}}>\] of real numbers satisfies the recurrence relation \[{{b}_{n+1}}=\frac{1}{3}\left( 2{{b}_{n}}+\frac{125}{{{b}^{2}}_{n}} \right),{{b}_{n}}\ne 0.\] Then find \[\underset{n\,\to \,\infty }{\mathop{\lim }}\,{{b}_{n}}.\]
  A)
  10 done clear
  B)
  15 done clear
  C)
  5 done clear
  D)
  25 done clear
  View Solution play_arrow

• question_answer12)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin [cosx]}{1+[cosx]}\] (\[[\,\cdot \,]\] denotes the greatest integer function)
  A)
  Equal to 1 done clear
  B)
  Equal to 0 done clear
  C)
  Does not exist done clear
  D)
  None of these done clear
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• question_answer13)
  If \[y=\left( 1+\frac{1}{x} \right)\left( 1+\frac{2}{x} \right)\left( 1+\frac{3}{x} \right)....\left( 1+\frac{n}{x} \right)\] and \[x\ne 0.\] then \[\frac{dy}{dx}\] when \[x=-1\] is
  A)
  \[n!\] done clear
  B)
  \[(n-1)!\] done clear
  C)
  \[{{(-1)}^{n}}(n-1)!\] done clear
  D)
  \[{{(-1)}^{n}}n!\] done clear
  View Solution play_arrow

• question_answer14)
  A triangle has two of its vertices at \[P(a,0),Q(0,b)\] and the third vertex \[R(x,\,\,y)\] is moving along the straight line \[y=x.\] If A be the area of the triangle. Then \[\frac{dA}{dx}\] is equal to
  A)
  \[\frac{a-b}{2}\] done clear
  B)
  \[\frac{a-b}{4}\] done clear
  C)
  \[=-\left( \frac{a+b}{2} \right)\] done clear
  D)
  \[\frac{a+b}{4}\] done clear
  View Solution play_arrow

• question_answer15)
  Derivative of \[{{\left( \sqrt{x}+\frac{1}{\sqrt{x}} \right)}^{2}}\] is
  A)
  \[\frac{1}{{{x}^{2}}}\] done clear
  B)
  \[1-\frac{1}{{{x}^{2}}}\] done clear
  C)
  1 done clear
  D)
  \[1+\frac{1}{{{x}^{2}}}\] done clear
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• question_answer16)
  If f be a function given by \[f(x)=2{{x}^{2}}+3x-5.\] Then, \[f'(0)=mf'(-1),\] where m is equal to
  A)
  -1 done clear
  B)
  -2 done clear
  C)
  -3 done clear
  D)
  -4 done clear
  View Solution play_arrow

• question_answer17)
  Let \[f(x)={{x}^{2}}-1,0<x<2\] and \[2x+3,2\le x<3.\] The quadratic equation whose roots are, \[\underset{x\to 2-0}{\mathop{\lim }}\,f(x)\] And \[\underset{x\to 2+\,0}{\mathop{\lim }}\,f(x)\] is
  A)
  \[{{x}^{2}}-6x+9=0\] done clear
  B)
  \[{{x}^{2}}-10x+21=0\] done clear
  C)
  \[{{x}^{2}}-14x+49=0\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer18)
  If \[{{x}_{1}}=3\] and \[{{x}_{n+1}}=\sqrt{2+{{x}_{n}},}n\ge 1,\] then \[\underset{n\,\to \,\infty }{\mathop{\lim }}\,{{x}_{n}}\] is equal to
  A)
  \[-1\] done clear
  B)
  \[2\] done clear
  C)
  \[\sqrt{5}\] done clear
  D)
  \[3\] done clear
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• question_answer19)
  Let \[f(x)=\left\{ \begin{matrix}    x\sin \left( \frac{1}{x} \right)+\sin \left( \frac{1}{{{x}^{2}}} \right),x\ne 0  \\    0,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x=0  \\ \end{matrix} \right.\]then \[\underset{x\to \infty }{\mathop{\lim }}\,f(x)\] equals
  A)
  0 done clear
  B)
  \[-1/2\] done clear
  C)
  1 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer20)
  Let \[f(x)=x-[x],\] where [x] denotes the greatest integer \[\le x\] and \[g(x)=\underset{n\,\to \,\infty }{\mathop{\lim }}\,\frac{{{\{f(x)\}}^{2n}}-1}{{{\{f(x)\}}^{2n}}+1},\] then g(x) is equal to
  A)
  0 done clear
  B)
  1 done clear
  C)
  -1 done clear
  D)
  None of these   done clear
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• question_answer21)
  \[\underset{x\to 0}{\mathop{\lim }}\,{{\left| x \right|}^{[cosx]}}\] is
  A)
  1 done clear
  B)
  Does not exist done clear
  C)
  0 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer22)
  Let\[f(x)=\underset{n\,\to \,\infty }{\mathop{\lim }}\,\frac{\log (2+x)-{{x}^{2n}}\sin x}{1+{{x}^{2n}}}\]. Then
  A)
  \[\underset{x\to {{1}^{+}}}{\mathop{\lim }}\,f(x)\ne \underset{x\to {{1}^{-}}}{\mathop{\lim }}\,f(x)\] done clear
  B)
  \[\underset{x\to {{1}^{+}}}{\mathop{\lim }}\,f(x)=sin1\] done clear
  C)
  \[\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,f(x)\] doesn?t exist done clear
  D)
  None of these done clear
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• question_answer23)
  If \[\frac{d}{dx}\left( \frac{1+{{x}^{4}}+{{x}^{8}}}{1+{{x}^{2}}+{{x}^{4}}} \right)=a{{x}^{3}}+bx,\] then
  A)
  \[a=4,b=2\] done clear
  B)
  \[a=4,b=-2\] done clear
  C)
  \[a=-2,b=4\] done clear
  D)
  None of these done clear
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• question_answer24)
  If \[f(x)={{\left( \frac{{{\sin }^{m}}x}{{{\sin }^{n}}x} \right)}^{m+n}}.{{\left( \frac{{{\sin }^{n}}x}{{{\sin }^{p}}x} \right)}^{n+p}}.{{\left( \frac{{{\sin }^{p}}\,x}{{{\sin }^{m}}x} \right)}^{p+m}}\] Then \[f'(x)\]is equal to
  A)
  0 done clear
  B)
  1 done clear
  C)
  \[{{\cos }^{m+n+px}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer25)
  If \[f(x)=\left\{ \begin{matrix}    \frac{{{[x]}^{2}}+\sin [x]}{[x]}for[x]\ne 0  \\    0for[x]=0  \\ \end{matrix} \right.\], where [x] denotes the greatest integer less than or equal to\[x,\] Then \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\] equals
  A)
  1 done clear
  B)
  0 done clear
  C)
  -1 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer26)
  Let \[f(x)=\sqrt{x-1}+\sqrt{x+24-10\sqrt{x-1}};\] \[1<x<26\] Be real valued function. Then \[f'(x)\]for \[1<x<26\] is
  A)
  0 done clear
  B)
  \[\frac{1}{\sqrt{x-1}}\] done clear
  C)
  \[2\sqrt{x-1}-5\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer27)
  If \[y=\frac{1}{1+{{x}^{\beta -\alpha }}+{{x}^{\gamma -\alpha }}}+\frac{1}{1+{{x}^{\alpha -\beta }}+{{x}^{\gamma -\beta }}}+\frac{1}{1+{{x}^{\alpha -\gamma }}+{{x}^{\beta -\gamma }}}\]then \[\frac{dy}{dx}\] is equal to
  A)
  0 done clear
  B)
  1 done clear
  C)
  \[(\alpha +\beta +\gamma ){{x}^{\alpha +\beta +\gamma -1}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer28)
  \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{{{n}^{p}}{{\sin }^{2}}(n!)}{n+1},0<p<1\] is equal to
  A)
  0 done clear
  B)
  \[\infty \] done clear
  C)
  1 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer29)
  Let \[f(x)=x{{(-1)}^{[1/x]}},x\ne 0,\] where [x] denotes the greatest integer less than or equal to x then, \[\underset{x\to 0}{\mathop{\lim }}\,f(x)=\]
  A)
  Does not exist done clear
  B)
  2 done clear
  C)
  0 done clear
  D)
  -1 done clear
  View Solution play_arrow

• question_answer30)
  \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin {{x}^{4}}-{{x}^{4}}\cos {{x}^{4}}+{{x}^{20}}}{{{x}^{4}}({{e}^{2{{x}^{4}}}}1-2{{x}^{4}})}\] is equal to
  A)
  0 done clear
  B)
  \[-1/6\] done clear
  C)
  \[1/6\] done clear
  D)
  Does not exist done clear
  View Solution play_arrow

• question_answer31)
  If \[f(x)=\frac{\sin ({{e}^{x-2}}-1)}{in(x-1)},\] then \[\underset{x\to 2}{\mathop{\lim }}\,f(x)\] is equal to
  A)
  \[-2\] done clear
  B)
  \[-1\] done clear
  C)
  \[0\] done clear
  D)
  1 done clear
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• question_answer32)
  The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{27}^{x}}-{{9}^{x}}-{{3}^{x}}+1}{\sqrt{2}-\sqrt{1+\cos x}}\] is
  A)
  \[4\sqrt{2}{{(log3)}^{2}}\] done clear
  B)
  \[8\sqrt{2}{{(log3)}^{2}}\] done clear
  C)
  \[2\sqrt{2}{{(log3)}^{2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer33)
  If \[f(x)=\sqrt{{{x}^{2}}-10x+25},\] then the derivative of f(x) on the interval \[[0,7]\] is
  A)
  1 done clear
  B)
  -1 done clear
  C)
  0 done clear
  D)
  None of these   done clear
  View Solution play_arrow

• question_answer34)
  Let f(x) be a polynomial function satisfying \[f(x).f\left( \frac{1}{x} \right)=f(x)+f\left( \frac{1}{x} \right).\] if \[f(4)=65\] and \[{{l}_{1}},{{l}_{2}},{{l}_{3}}\]are in \[GP,\] then \[f'({{l}_{1}}),f'({{l}_{2}}),f'({{l}_{3}})\] are in
  A)
  AP done clear
  B)
  GP done clear
  C)
  HP done clear
  D)
  None of these done clear
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• question_answer35)
  The value of \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{({{4}^{x}}-1)}^{3}}}{\sin \frac{{{x}^{2}}}{4}\log (1+3x)},\] is
  A)
  \[\frac{4}{3}{{(in4)}^{2}}\] done clear
  B)
  \[\frac{4}{3}{{(In4)}^{3}}\] done clear
  C)
  \[\frac{3}{2}{{(In4)}^{2}}\] done clear
  D)
  \[\frac{3}{2}{{(In4)}^{3}}\] done clear
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• question_answer36)
  If \[{{A}_{i}}=\frac{x-{{a}_{i}}}{\left| x-{{a}_{i}} \right|},i=1,2,3,....,n\] and  \[{{a}_{1}}<{{a}_{2}}<{{a}_{3}}....{{a}_{n}},\]then \[\underset{x\to {{a}_{m}}}{\mathop{\lim }}\,({{A}_{1}}{{A}_{2}}...{{A}_{n}}),1\le m\le n\]
  A)
  Is equal to \[{{(-1)}^{m}}\] done clear
  B)
  Is equal to \[{{(-1)}^{m+1}}\] done clear
  C)
  Is equal to \[{{(-1)}^{m-1}}\] done clear
  D)
  Does not exist done clear
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• question_answer37)
  If [.] denotes the greatest integer function, then \[\underset{n\,\to \,\infty }{\mathop{\lim }}\,\frac{[x]+[2x]+...+[nx]}{{{n}^{2}}}\] is
  A)
  0 done clear
  B)
  \[x\] done clear
  C)
  \[\frac{x}{2}\] done clear
  D)
  \[\frac{{{x}^{2}}}{2}\] done clear
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• question_answer38)
  The limit \[\underset{n\,\to \,\infty }{\mathop{\lim }}\,\,\underset{r=3}{\overset{n}{\mathop{\prod }}}\,\,\,\frac{{{r}^{3}}-8}{{{r}^{3}}+8}\] is equal to
  A)
  \[\frac{2}{7}\] done clear
  B)
  \[\frac{1}{12}\] done clear
  C)
  \[\frac{19}{52}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer39)
  If \[m,n\in {{I}_{0}}\] and \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\tan 2x-n\sin x}{{{x}^{3}}}=\]some integer, then value of this limit is
  A)
  3 done clear
  B)
  2 done clear
  C)
  \[\frac{16+n}{12}\] done clear
  D)
  None of these done clear
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• question_answer40)
  \[\underset{x\to \infty }{\mathop{\lim }}\,{{\left( \frac{{{x}^{2}}+5x+3}{{{x}^{2}}+x+2} \right)}^{x}}\]
  A)
  \[{{e}^{4}}\] done clear
  B)
  \[{{e}^{2}}\] done clear
  C)
  \[{{e}^{3}}\] done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer41)
  \[\underset{x\to \pi /2}{\mathop{\lim }}\,\frac{\left[ \frac{x}{2} \right]}{ln\,(sin\,x)}\] (where [.] denotes the greatest integer function)
  A)
  Does not exist done clear
  B)
  Equals 1 done clear
  C)
  Equals 0 done clear
  D)
  Equals -1 done clear
  View Solution play_arrow

• question_answer42)
  \[\underset{x\to \infty }{\mathop{\lim }}\,\left( \frac{{{x}^{100}}}{{{e}^{x}}}+{{\left( \cos \frac{2}{x} \right)}^{{{x}^{2}}}} \right)=\]
  A)
  \[{{e}^{-1}}\] done clear
  B)
  \[{{e}^{-4}}\] done clear
  C)
  \[(1+{{e}^{-2}})\] done clear
  D)
  \[{{e}^{-2}}\] done clear
  View Solution play_arrow

• question_answer43)
  If \[a=\min \{{{x}^{2}}+4x+5,x\in R\}\]and \[b=\underset{\theta \to 0}{\mathop{\lim }}\,\frac{1-\cos 2\theta }{{{\theta }^{2}}},\] then the value of \[\sum\limits_{r=0}^{n}{{{a}^{r}}.{{b}^{n-r}}}\] is
  A)
  \[\frac{{{2}^{n+1}}-1}{{{4.2}^{n}}}\] done clear
  B)
  \[{{2}^{n+1}}-1\] done clear
  C)
  \[\frac{{{2}^{n+1}}-1}{{{3.2}^{n}}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer44)
  \[\underset{x\to 0}{\mathop{\lim }}\,\left[ \frac{\sin [x-3]}{[x-3]} \right],\] where \[[.]\] denotes greatest integer function is
  A)
  0 done clear
  B)
  1 done clear
  C)
  Does not exist done clear
  D)
  sin 1 done clear
  View Solution play_arrow

• question_answer45)
  If \[\underset{x\to 0}{\mathop{\lim }}\,\frac{(sinnx)[(a-n)nx-tanx]}{{{x}^{2}}}=0,\] then the value of a
  A)
  \[\frac{1}{n}\] done clear
  B)
  \[n-\frac{1}{n}\] done clear
  C)
  \[n+\frac{1}{n}\] done clear
  D)
  None done clear
  View Solution play_arrow

• question_answer46)
  \[\underset{x\to 0}{\mathop{\lim }}\,\left[ \cos e{{c}^{3}}x.\cot x-2{{\cot }^{3}}x.\cos ecx+\frac{{{\cot }^{4}}x}{\sec x} \right]\] is equal to
  A)
  1 done clear
  B)
  -1 done clear
  C)
  0 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer47)
  \[\underset{x\to \infty }{\mathop{\lim }}\,\left( \frac{{{x}^{2}}}{3x-2}-\frac{x}{3} \right)=\]
  A)
  \[\frac{1}{3}\] done clear
  B)
  \[\frac{2}{3}\] done clear
  C)
  \[\frac{-2}{3}\] done clear
  D)
  \[\frac{2}{9}\] done clear
  View Solution play_arrow

• question_answer48)
  The value of \[\underset{x\to \infty }{\mathop{\lim }}\,\frac{({{2}^{{{x}^{n}}}}){{e}^{\frac{1}{^{x}}}}-({{3}^{{{x}^{n}}}}){{e}^{\frac{1}{x}}}}{{{x}^{n}}}\](where \[n\in N\]) is
  A)
  \[\log n\left( \frac{2}{3} \right)\] done clear
  B)
  0 done clear
  C)
  \[n\log n\left( \frac{2}{3} \right)\] done clear
  D)
  Not defined done clear
  View Solution play_arrow

• question_answer49)
  \[\underset{x\to 1}{\mathop{\lim }}\,\frac{(1-x)(1-{{x}^{2}})...(1-{{x}^{2n}})}{{{\{(1-x)(1-{{x}^{2}})...(1-{{x}^{n}})\}}^{2}}},n\in N,\] equals
  A)
  \[^{2n}{{P}_{n}}\] done clear
  B)
  \[^{2n}{{C}_{n}}\] done clear
  C)
  \[(2n)!\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer50)
  The value of  \[\underset{x\to \frac{\pi }{2}}{\mathop{\lim }}\,{{\left[ {{1}^{1/{{\cos }^{2}}x}}+{{2}^{1/{{\cos }^{2}}x}}+...+{{n}^{1/co{{s}^{2}}x}} \right]}^{{{\cos }^{2x}}}}\] is
  A)
  0 done clear
  B)
  n done clear
  C)
  \[\infty \] done clear
  D)
  \[\frac{n(n+1)}{2}\] done clear
  View Solution play_arrow

• question_answer51)
  The value of \[\underset{x\to 0}{\mathop{\lim }}\,{{\log }_{e}}{{(sinx)}^{\tan x}}\] is
  A)
  1 done clear
  B)
  -1 done clear
  C)
  0 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer52)
  \[\underset{x\to 0}{\mathop{\lim }}\,\left( \frac{10\sin 9x}{9\sin 10x} \right)\left( \frac{8\sin 7x}{7\sin 8x} \right)\left( \frac{6\sin 5x}{5\sin 6x} \right)\left( \frac{4\sin 3x}{3\sin 4x} \right)\] \[\left( \frac{\sin x}{\sin 2x} \right)=\]
  A)
  \[\frac{63}{256}\] done clear
  B)
  \[\frac{1}{6}\] done clear
  C)
  \[\frac{6}{5}\] done clear
  D)
  \[\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer53)
  What is \[\underset{x\to 0}{\mathop{\lim }}\,\frac{x}{\sqrt{1-\cos x}}\] equal to?
  A)
  \[\sqrt{2}\] done clear
  B)
  \[-\sqrt{2}\] done clear
  C)
  \[\frac{1}{\sqrt{2}}\] done clear
  D)
  Limit does not exist done clear
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• question_answer54)
  Let \[f(x)=4\] and \[f'(x)=4.\]Then \[\underset{x\to 2}{\mathop{\lim }}\,\frac{xf(2)-2f(x)}{x-2}\] is given by
  A)
  2 done clear
  B)
  -2 done clear
  C)
  -4 done clear
  D)
  3 done clear
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• question_answer55)
  \[\underset{x\to \frac{{{\pi }^{-}}}{2}}{\mathop{\lim }}\,{{[1+{{(cos\,x)}^{\cos x}}]}^{2}}\] is equal to
  A)
  Does not exist done clear
  B)
  1 done clear
  C)
  e done clear
  D)
  4 done clear
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• question_answer56)
  Let \[\alpha \] and \[\beta \] be the roots of \[a{{x}^{2}}+bx+c=0.\]Then \[\underset{x\to \alpha }{\mathop{\lim }}\,\frac{1-\cos (a{{x}^{2}}+bx+c)}{{{(x-\alpha )}^{2}}}\] is equal to:
  A)
  0 done clear
  B)
  \[\frac{1}{2}{{(\alpha -\beta )}^{2}}\] done clear
  C)
  \[\frac{{{a}^{2}}}{2}{{(\alpha -\beta )}^{2}}\] done clear
  D)
  None of these done clear
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• question_answer57)
  If \[f(x)=\underset{n\,\to \,\infty }{\mathop{\lim }}\,n({{x}^{1/n}}-1),\] then for \[x>0,\,\,y>0,\]\[f(xy)\] is equal to
  A)
  \[f(x)f(y)\] done clear
  B)
  \[f(x)+f(y)\] done clear
  C)
  \[f(x)-f(y)\] done clear
  D)
  None of these done clear
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• question_answer58)
  Let \[f:R\to R\] be such that \[f(1)=3\] and \[f'(1)=6.\] Then \[\underset{x\to 0}{\mathop{\lim }}\,{{\left( \frac{f(1+x)}{f(1)} \right)}^{1/x}}\] equals
  A)
  1 done clear
  B)
  \[{{e}^{1/2}}\] done clear
  C)
  \[{{e}^{2}}\] done clear
  D)
  \[{{e}^{3}}\] done clear
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• question_answer59)
  If \[m,\,\,\,n\in {{I}_{0}}\] and \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\tan 2x-n\sin x}{{{x}^{3}}}=\] some integer, then value of this limit is
  A)
  3 done clear
  B)
  2 done clear
  C)
  \[\frac{16+n}{12}\] done clear
  D)
  None of these done clear
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• question_answer60)
  If \[{{z}_{r}}=\cos \frac{r\alpha }{{{n}^{2}}}+i\sin \frac{r\alpha }{{{n}^{2}}},\] where \[r=1,2,3,...n,\] then \[\underset{x\to \infty }{\mathop{\lim }}\,{{z}_{1}}{{z}_{2}}{{z}_{3}}...{{z}_{n}}\] is equal to
  A)
  \[\cos \alpha +i\sin \alpha \] done clear
  B)
  \[\cos (\alpha /2)-i\,\,sin(\alpha /2)\] done clear
  C)
  \[{{e}^{i\alpha /2}}\] done clear
  D)
  \[\sqrt[3]{{{e}^{i\alpha }}}\] done clear
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• question_answer61)
  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
  A)
  \[a=1\] and \[b=2\] done clear
  B)
  \[a=1,b\in R\] done clear
  C)
  \[a\in R,b=2\] done clear
  D)
  \[a\in R,b\in R\] done clear
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• question_answer62)
  The limit \[\underset{x\to 0}{\mathop{\lim }}\,{{(cosx)}^{1/\sin x\frac{1}{\sin x}}}\] is equal to
  A)
  \[e\] done clear
  B)
  \[{{e}^{-1}}\] done clear
  C)
  1 done clear
  D)
  Does not exist done clear
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• question_answer63)
  \[\underset{x\to 0}{\mathop{\lim }}\,{{\left\{ \frac{1+\operatorname{tanx}}{1+\sin x} \right\}}^{\cos ecx}}\] is equal to
  A)
  \[\frac{1}{e}\] done clear
  B)
  1 done clear
  C)
  \[e\] done clear
  D)
  \[{{e}^{2}}\] done clear
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• question_answer64)
  Evaluate \[\underset{x\to \infty }{\mathop{\lim }}\,{{2}^{x-1}}\tan \left( \frac{a}{{{2}^{x}}} \right).\]
  A)
  \[a\] done clear
  B)
  \[2a\] done clear
  C)
  \[\frac{a}{2}\] done clear
  D)
  \[4a\] done clear
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• question_answer65)
  What is \[\underset{n\to \infty }{\mathop{\lim }}\,\frac{1+2+3+...+n}{{{1}^{2}}+{{2}^{2}}+{{3}^{2}}+...{{n}^{2}}}\] equal to?
  A)
  5 done clear
  B)
  2 done clear
  C)
  1 done clear
  D)
  0 done clear
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• question_answer66)
  What is \[\underset{x\to 0}{\mathop{\lim }}\,\frac{\sin 2x+4x}{2x+\sin 4x}\] equal to?
  A)
  0 done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  1 done clear
  D)
  2 done clear
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• question_answer67)
  If \[\underset{x\to 0}{\mathop{\lim }}\,{{(1+asinx)}^{\cos ecx}}=3.\]then a is
  A)
  ln 2 done clear
  B)
  ln 3 done clear
  C)
  ln 4 done clear
  D)
  \[{{e}^{3}}\] done clear
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• question_answer68)
  The value of \[\underset{x\to \pi /2}{\mathop{\lim }}\,{{\tan }^{2}}x(\sqrt{2{{\sin }^{2}}x+3\sin x+4}\] \[-\sqrt{{{\sin }^{2}}x+6\sin x+2)}\] is equal to
  A)
  \[\frac{1}{10}\] done clear
  B)
  \[\frac{1}{11}\] done clear
  C)
  \[\frac{1}{12}\] done clear
  D)
  \[\frac{1}{8}\] done clear
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• question_answer69)
  If \[\underset{x\to a}{\mathop{\lim }}\,\left[ \frac{f(x)}{g(x)} \right]\]exist, then which one of the following correct?
  A)
  Both \[\underset{x\to a}{\mathop{\lim }}\,f(x)\] and \[\underset{x\to a}{\mathop{\lim }}\,g(x)\] must exist done clear
  B)
  \[\underset{x\to a}{\mathop{\lim }}\,f(x)\] need not exist but \[\underset{x\to a}{\mathop{\lim }}\,g(x)\] must exist done clear
  C)
  Both \[\underset{x\to a}{\mathop{\lim }}\,f(x)\] and \[\underset{x\to a}{\mathop{\lim }}\,g(x)\] need not exist done clear
  D)
  None of these done clear
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• question_answer70)
  The value of \[\underset{n\to \infty }{\mathop{\lim }}\,\left[ \sqrt[3]{{{(n+1)}^{2}}}-\sqrt[3]{{{(n-1)}^{2}}} \right]\] is
  A)
  1 done clear
  B)
  -1 done clear
  C)
  0 done clear
  D)
  \[-\infty \] done clear
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