JEE Main & Advanced Mathematics Functions Question Bank

done Continuity Total Questions - 80

• question_answer1)
  If\[f(x)=|x-2|\], then [Roorkee 1984]
  A)
             \[\underset{x\to 2+}{\mathop{\lim }}\,f(x)\ne 0\]                      done clear
  B)
             \[\underset{x\to 2-}{\mathop{\lim }}\,f(x)\ne 0\] done clear
  C)
             \[\underset{x\to 2+}{\mathop{\lim }}\,f(x)\ne \underset{x\to 2-}{\mathop{\lim }}\,f(x)\]    done clear
  D)
             \[f(x)\]is continuous at \[x=2\] done clear
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• question_answer2)
  If the function \[f(x)=\left\{ \begin{align}   & \frac{k\cos x}{\pi -2x},\text{when }x\ne \frac{\pi }{2} \\  & 3,\ \ \ \ \ \ \ \ \ \text{when }x=\frac{\pi }{2} \\ \end{align} \right.\] be continuous at \[x=\frac{\pi }{2}\], then k =      
  A)
             3 done clear
  B)
             6 done clear
  C)
             12 done clear
  D)
             None of these done clear
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• question_answer3)
  The function \[f(x)=\frac{\log (1+ax)-\log (1-bx)}{x}\]is not defined at \[x=0\]. The value which should be assigned to f at x =0 so that it is continuos at \[x=0\], is [IIT 1983; MP PET 1995; Karnataka CET 1999; Kurukshetra CEE 2002; AMU 2002]
  A)
             \[a-b\] done clear
  B)
             \[a+b\] done clear
  C)
             \[\log a+\log b\] done clear
  D)
             \[\log a-\log b\] done clear
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• question_answer4)
  Let \[f(x)=\left\{ \begin{align}   & \frac{{{x}^{3}}+{{x}^{2}}-16x+20}{{{(x-2)}^{2}}},\text{if}\ x\ne 2 \\  & \ \ \ \ \ \,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \,\ k,\ \text{if}\ x=2 \\ \end{align} \right.\].  If \[f(x)\] be continuous for all x, then k =                                        [IIT 1981]
  A)
             7 done clear
  B)
             ?7 done clear
  C)
             \[\pm 7\] done clear
  D)
             None of these done clear
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• question_answer5)
  Let \[f(x)=\left\{ \begin{align}   & {{x}^{2}}+k,\ \ \ \ \text{when}\ \ x\ge 0 \\  & -{{x}^{2}}-k,\ \ \text{when }x<0 \\ \end{align} \right.\]. If the function\[f(x)\] be continuous at \[x=0\], then k =
  A)
             0 done clear
  B)
             1 done clear
  C)
             2 done clear
  D)
             ?2 done clear
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• question_answer6)
  In order that the function\[f(x)={{(x+1)}^{1/x}}\]is continuous at \[x=0\], \[f(0)\]must be defined as              [MNR 1989]
  A)
             \[f(0)=0\] done clear
  B)
             \[f(0)=e\] done clear
  C)
             \[f(0)=1/e\] done clear
  D)
             \[f(0)=1\] done clear
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• question_answer7)
  If \[f(x)=\left\{ \begin{align}   & x,\ \ \text{when}0<x<1/2 \\  & 1,\ \ \ \text{when }x=1/2 \\  & 1-x,\text{when}\ \text{1/2}<x<\text{1} \\ \end{align} \right.\], then
  A)
             \[\underset{x\to 1/2+}{\mathop{\lim }}\,f(x)=2\]                       done clear
  B)
             \[\underset{x\to 1/2-}{\mathop{\lim }}\,f(x)=2\] done clear
  C)
             \[f(x)\]is continuous at \[x=\frac{1}{2}\]                                     done clear
  D)
             \[f(x)\]is discontinuous at \[x=\frac{1}{2}\] done clear
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• question_answer8)
  If \[f(x)=\left\{ \begin{align}   & ({{x}^{2}}/a)-a,\ \ \text{when}\ x<a \\  & \ \ \ \ \ \ \ \ \ \ \ 0,\ \ \text{when}\ x=a\text{,} \\  & a-({{x}^{2}}/a),\ \ \text{when }x>a \\ \end{align} \right.\] then
  A)
             \[\underset{x\to a}{\mathop{\lim }}\,f(x)=a\] done clear
  B)
             \[f(x)\]is continuous at\[x=a\] done clear
  C)
             \[f(x)\]is discontinuous at\[x=a\]     done clear
  D)
             None of these done clear
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• question_answer9)
  If\[f(x)=\left\{ \begin{align}   & {{e}^{1/x}},\ \text{when}\ x\ne 0 \\  & 0,\ \ \ \ \ \text{when}\ x=0 \\ \end{align} \right.\], then                   
  A)
             \[\underset{x\to 0+}{\mathop{\lim }}\,f(x)=e\] done clear
  B)
             \[\underset{x\to 0+}{\mathop{\lim }}\,f(x)=0\] done clear
  C)
             \[f(x)\]is discontinuous at \[x=0\] done clear
  D)
             None of these done clear
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• question_answer10)
  If \[f(x)=\left\{ \begin{align}   & \frac{{{x}^{2}}-4x+3}{{{x}^{2}}-1},\ \text{for}\ x\ne 1 \\  & \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2,\ \text{for }x=1 \\ \end{align} \right.\], then [IIT 1972]
  A)
             \[\underset{x\to 1+}{\mathop{\lim }}\,f(x)=2\] done clear
  B)
             \[\underset{x\to 1-}{\mathop{\lim }}\,f(x)=3\] done clear
  C)
             \[f(x)\]is discontinuous at \[x=1\] done clear
  D)
             None of these done clear
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• question_answer11)
  The points at which the function \[f(x)=\frac{x+1}{{{x}^{2}}+x-12}\] is discontinuous, are
  A)
             ?3, 4 done clear
  B)
             3, ?4 done clear
  C)
             ?1,?3, 4 done clear
  D)
             ?1, 3, 4 done clear
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• question_answer12)
  If \[f(x)=\left\{ \begin{matrix}    {{e}^{x}}+ax, & x<0  \\    b{{(x-1)}^{2}}, & x\ge 0  \\ \end{matrix} \right.\] then [DSSE 1986]
  A)
             \[\underset{x\to 0+}{\mathop{\lim }}\,f(x)\ne 2\] done clear
  B)
             \[\underset{x\to 0-}{\mathop{\lim }}\,f(x)=0\] done clear
  C)
             \[f(x)\]is continuous at\[x=0\] done clear
  D)
             None of these done clear
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• question_answer13)
  If \[f(x)=\left\{ \begin{align}   & {{x}^{2}}\sin \frac{1}{x},\ \ \ \text{when }x\ne 0 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,0,\,\,\,\,\text{when}\,x=0 \\ \end{align} \right.\], then
  A)
             \[f(0+0)=1\] done clear
  B)
             \[f(0-0)=1\] done clear
  C)
             f is continuous at\[x=0\] done clear
  D)
             None of these done clear
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• question_answer14)
  The value of k so that the function \[f(x)=\left\{ \begin{align}   & k(2x-{{x}^{2}}),\ \ \ \text{when}\,x<0 \\  & \,\,\,\,\,\,\,\,\,\cos x,\,\,\,\,\,\,\text{when}\,x\ge \text{0} \\ \end{align} \right.\]is continuous at\[x=0\], is                                      
  A)
             1 done clear
  B)
             2 done clear
  C)
             4 done clear
  D)
             None of these done clear
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• question_answer15)
  If \[f(x)=\left\{ \begin{align}   & \frac{x}{{{e}^{1/x}}+1},\,\,\text{when}\,\,x\ne 0 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,0,\,\,\text{when }x=0 \\ \end{align} \right.\], then
  A)
             \[\underset{x\to 0+}{\mathop{\lim }}\,f(x)=1\] done clear
  B)
             \[\underset{x\to 0-}{\mathop{\lim }}\,f(x)=1\] done clear
  C)
             \[f(x)\]is continuous at\[x=0\] done clear
  D)
             None of these done clear
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• question_answer16)
  If \[f(x)=\left\{ \begin{align}   & {{(1+2x)}^{1/x}},\,\text{for }x\ne 0 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{e}^{2}},\,\text{for }x=0\,\,\, \\ \end{align} \right.\], then
  A)
    \[\underset{x\to 0+}{\mathop{\lim }}\,f(x)=e\]                                    done clear
  B)
             \[\underset{x\to 0-}{\mathop{\lim }}\,f(x)={{e}^{2}}\] done clear
  C)
             \[f(x)\]is discontinuous at \[x=0\]    done clear
  D)
             None of these done clear
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• question_answer17)
  If \[f(x)=\left\{ \begin{align}   & {{2}^{1/x}},\text{for}\,x\ne 0 \\  & \,\,\,\,\,\,\,3,\text{for}\,x=\text{0} \\ \end{align} \right.\], then
  A)
             \[\underset{x\to 0+}{\mathop{\lim }}\,f(x)=0\] done clear
  B)
             \[\underset{x\to 0-}{\mathop{\lim }}\,f(x)=\infty \] done clear
  C)
             \[f(x)\]is continuous at\[x=0\] done clear
  D)
   None of these done clear
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• question_answer18)
  If \[f(x)=\left\{ \begin{align}   & \frac{1}{x}\sin {{x}^{2}},\,x\ne 0 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,0,\,x=0 \\ \end{align} \right.\], then
  A)
             \[\underset{x\to 0+}{\mathop{\lim }}\,f(x)\ne 0\] done clear
  B)
             \[\underset{x\to 0-}{\mathop{\lim }}\,f(x)\ne 0\] done clear
  C)
   f(x) is continuous at\[x=0\] done clear
  D)
             None of these done clear
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• question_answer19)
  If \[f(x)=\left\{ \begin{align}   & x-1,\,\,\,x<0 \\  & \,\,\,\,\,\,\frac{1}{4},\,\,x=0 \\  & \,\,\,\,\,\,\,{{x}^{2}},\,\,x>0 \\ \end{align} \right.\], then [Roorkee 1988]
  A)
             \[\underset{x\to 0+}{\mathop{\lim }}\,f(x)=1\] done clear
  B)
             \[\underset{x\to 0-}{\mathop{\lim }}\,f(x)=1\] done clear
  C)
             \[f(x)\]is discontinuous at\[x=0\]     done clear
  D)
             None of these done clear
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• question_answer20)
  Which of the following statements is true for graph \[f(x)=\log x\]     
  A)
             Graph shows that function is continuous done clear
  B)
             Graph shows that function is discontinuous done clear
  C)
             Graph finds for negative and positive values of x done clear
  D)
             Graph is symmetric along x-axis done clear
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• question_answer21)
  If function \[f(x)=\left\{ \begin{align}   & \frac{{{x}^{2}}-1}{x-1},\,\,\text{when}\,\,x\ne 1 \\  & \,\,\,\,\,\,\,\,\,\,\,\,k,\,\text{when}\,\,x=1 \\ \end{align} \right.\]is continuous at \[x=1\], then the value of k will be            
  A)
             ?1 done clear
  B)
             2 done clear
  C)
             ?3 done clear
  D)
             ?2 done clear
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• question_answer22)
  At which points the function\[f(x)=\frac{x}{[x]}\], where\[[.]\] is greatest integer function, is discontinuous
  A)
             Only positive integers                done clear
  B)
             All positive and negative integers and (0, 1) done clear
  C)
             All rational numbers done clear
  D)
             None of these done clear
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• question_answer23)
  For the function \[f(x)=\left\{ \begin{align}   & \frac{{{\sin }^{2}}ax}{{{x}^{2}}},\,\text{when}\,x\ne 0 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,1,\text{when}\,x=0 \\ \end{align} \right.\] which one is a true statement                              
  A)
             \[f(x)\]is continuous at \[x=0\] done clear
  B)
             \[f(x)\]is discontinuous at \[x=0\], when \[a\ne \pm 1\] done clear
  C)
             \[\underset{x\to 1}{\mathop{\lim }}\,(1-x+[x-1]+[1-x])\] is continuous at \[x=a\] done clear
  D)
             None of these done clear
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• question_answer24)
  If \[f(x)=\left\{ \begin{align}   & \,\,\,\,\,\,\,\,\,-{{x}^{2}},\,\text{when }x\le 0 \\  & \,\,\,\,\,5x-4,\,\text{when}0<x\le 1 \\  & 4{{x}^{2}}-3x,\,\text{when }1<x<2 \\  & \,\,\,\,\,3x+4,\text{when }x\ge 2 \\ \end{align} \right.\], then
  A)
             \[f:R\to R\]is continuous at \[x=0\] done clear
  B)
             \[f(x)\] is continuous \[x=2\] done clear
  C)
             \[f(x)\]is discontinuous at\[x=1\] done clear
  D)
             None of these done clear
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• question_answer25)
  If \[f(x)=\left\{ \begin{align}   & {{\sin }^{-1}}|x|,\text{when}\,x\ne 0 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,0,\,\text{when }x=0 \\ \end{align} \right.\] then
  A)
             \[\underset{x\to 0+}{\mathop{\lim }}\,f(x)\ne 0\]                      done clear
  B)
             \[\underset{x\to 0-}{\mathop{\lim }}\,f(x)\ne 0\] done clear
  C)
             \[f(x)\]is continuous at\[x=0\]  done clear
  D)
             None of these done clear
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• question_answer26)
  If \[f(x)=\left\{ \begin{align}   & \frac{\sin 2x}{5x},\text{when}\,x\ne 0 \\  & \,\,\,\,\,\,\,\,\,\,\,\,k,\text{when }x=0 \\ \end{align} \right.\] is continuous at\[x=0\], then the value of k will be [AI CBSE 1991]
  A)
             1 done clear
  B)
             \[\frac{2}{5}\] done clear
  C)
             \[-\frac{2}{5}\] done clear
  D)
             None of these done clear
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• question_answer27)
  If \[f(x)=\left\{ \begin{align}   & 1+{{x}^{2}},\,\,\,\text{when}\,0\le x\le 1 \\  & 1-x\,\,\,,\text{when}\,\,x>1 \\ \end{align} \right.\],  then
  A)
             \[\underset{x\to {{1}^{+}}}{\mathop{\lim }}\,f(x)\ne 0\]            done clear
  B)
   \[\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,f(x)\ne 2\] done clear
  C)
   \[f(x)\]is discontinuous at \[x=1\] done clear
  D)
             None of these done clear
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• question_answer28)
  If \[f(x)=\left\{ \begin{align}   & \frac{{{x}^{2}}-1}{x+1},\,\text{when }x\ne -1 \\  & \,\,\,\,\,\,\,\,-2,\,\text{when }x=-1 \\ \end{align} \right.\],then
  A)
             \[\underset{x\to {{(-1)}^{-}}}{\mathop{\lim }}\,f(x)=-2\] done clear
  B)
             \[\underset{x\to {{(-1)}^{+}}}{\mathop{\lim }}\,f(x)=-2\] done clear
  C)
             \[f(x)\]is continuous at \[x=-1\] done clear
  D)
             All the above are correct done clear
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• question_answer29)
  If \[f(x)=\left\{ \begin{align}   & \frac{5}{2}-x\,,\,\text{when}\,x<2 \\  & \,\,\,1\,\,\,\,\,\,,\,\text{when }x=2 \\  & x-\frac{3}{2},\text{when}\,x>2 \\ \end{align} \right.\], then
  A)
             \[f(x)\]is continuous at \[x=2\] done clear
  B)
             \[f(x)\]is discontinuous at \[x=2\] done clear
  C)
             \[\underset{x\to 2}{\mathop{\lim }}\,f(x)=1\] done clear
  D)
             None of these done clear
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• question_answer30)
  If \[f(x)=|x-b|,\]then function                                          [AI CBSE 1984]
  A)
             Is continuous at \[x=1\] done clear
  B)
             Is continuous at \[x=b\] done clear
  C)
             Is discontinuous at\[x=b\] done clear
  D)
             None of these done clear
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• question_answer31)
  If\[f(x)=\left\{ \begin{align}   & \frac{|x-a|}{x-a},\text{when}\,x\ne a \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,1,\text{when}\,x=a \\ \end{align} \right.\],then                             [AI CBSE 1983]
  A)
             \[f(x)\]is continuous at \[x=a\] done clear
  B)
             \[f(x)\]is discontinuous at \[x=a\] done clear
  C)
             \[\underset{x\to a}{\mathop{\lim }}\,f(x)=1\] done clear
  D)
             None of these done clear
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• question_answer32)
  If \[f(x)=\left\{ \begin{align}   & {{x}^{2}},\,\,\text{when}\,\,\,x\ne 1 \\  & \,\,\,2,\text{when}\,\,x=1 \\ \end{align} \right.\]then
  A)
             \[\underset{x\to 1}{\mathop{\lim }}\,f(x)=2\] done clear
  B)
             \[f(x)\]is continuous at \[x=1\] done clear
  C)
             \[f(x)\]is discontinuous at \[x=1\]    done clear
  D)
             None of these done clear
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• question_answer33)
  If \[f(x)=\left\{ \begin{align}   & 1+x,\ \text{when }x\le 2 \\  & 5-x,\,\text{when }\,x\le 3 \\ \end{align} \right.\], then
  A)
             \[f(x)\]is continuous at \[x=2\] done clear
  B)
             \[f(x)\]is discontinuous at \[A=0,\,B=1\] done clear
  C)
             \[f(x)\]is continuous at\[x=3\]  done clear
  D)
             None of these done clear
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• question_answer34)
  If\[f(x)=\left\{ \begin{align}   & \,\,\,\,\,\,\,\,\,\,\,\,\,\,1,\,\text{when}\,\,0<x\le \frac{3\pi }{4} \\  & 2\sin \frac{2}{9}x,\text{when}\,\frac{3\pi }{4}<x<\pi  \\ \end{align} \right.\], then                    [IIT 1991]
  A)
             \[f(x)\]is continuous at \[x=0\] done clear
  B)
             \[f(x)\]is continuous at \[x=\pi \] done clear
  C)
             \[f(x)\]is continuous at \[x=\frac{3\pi }{4}\] done clear
  D)
             \[f(x)\]is discontinuous at \[x=\frac{3\pi }{4}\] done clear
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• question_answer35)
  If \[f(x)=\left\{ \begin{align}   & \,\,\,\,\,\,\,\,\,\,\,\,x\sin x,\,\text{when }0<x\le \frac{\pi }{2} \\  & \frac{\pi }{2}\sin (\pi +x),\text{when}\frac{\pi }{\text{2}}<x<\pi  \\ \end{align} \right.\], then                    [IIT 1991]
  A)
             \[f(x)\]is discontinuous at \[x=\pi /2\] done clear
  B)
             \[f(x)\]is continuous at \[x=\pi /2\] done clear
  C)
             \[f(x)\]is continuous at \[x=0\] done clear
  D)
             None of these done clear
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• question_answer36)
  If \[f(x)=\left\{ \begin{align}   & \,\,\,\,\,\,\,\,\,\frac{1-\cos 4x}{{{x}^{2}}},\ \ \text{when}\,x<0 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,a,\,\,\,\text{when}\,\,x=0 \\  & \frac{\sqrt{x}}{\sqrt{(16+\sqrt{x})}-4},\,\,\text{when}\,x>0 \\ \end{align} \right.\],            is continuous at \[x=0\], then the value of 'a' will be                                                                                 [IIT 1990; AMU 2000]
  A)
             8 done clear
  B)
             ?8 done clear
  C)
             4 done clear
  D)
             None of these done clear
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• question_answer37)
  If \[f(x)=\left\{ \begin{align}   & a{{x}^{2}}-b,\,\,\text{when }0\le x<1 \\  & \,\,\,\,\,\,\,\,\,\,\,\,2,\text{when }x=1 \\  & \,\,\,\,x+1,\,\,\text{when1}<x\le 2 \\ \end{align} \right.\]is continuous at \[x=1\], then the most suitable value of a, b are       [BIT Ranchi 1983]
  A)
             \[a=2,\ b=0\] done clear
  B)
             \[a=1,\ b=-1\] done clear
  C)
             \[a=4,\ b=2\] done clear
  D)
             All the above done clear
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• question_answer38)
  If \[f(x)=\left\{ \begin{align}   & \frac{x-|x|}{x},\text{when}\,x\ne 0 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,2,\,\text{when}\,x=0 \\ \end{align} \right.\], then                    [AI CBSE 1982]
  A)
             \[f(x)\]is continuous at \[x=0\] done clear
  B)
             \[\left[ 0,\frac{\pi }{2} \right]\]is discontinuous at \[x=0\] done clear
  C)
             \[\underset{x\to 0}{\mathop{\lim }}\,f(x)=2\]                            done clear
  D)
             None of these done clear
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• question_answer39)
  If \[f(x)=\left\{ \begin{align}   & \frac{{{x}^{4}}-16}{x-2},\,\,\text{when}\,x\ne 2 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,16,\,\text{when}\,x=2 \\ \end{align} \right.\], then                    [AISSE 1984]
  A)
             \[f(x)\]is continuous at \[x=2\] done clear
  B)
             \[f(x)\]is discountinous at \[x=2\] done clear
  C)
             \[\underset{x\to 2}{\mathop{\lim }}\,f(x)=16\] done clear
  D)
             None of these done clear
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• question_answer40)
  If \[f(x)=\left\{ \begin{align}   & \,\,\,\,{{x}^{2}},\,\text{when}\,x\le 1 \\  & x+5,\text{when }x>\text{1} \\ \end{align} \right.\], then [AISSE 1983]
  A)
             \[f(x)\]is continuous at \[x=1\] done clear
  B)
             \[f(x)\]is discontinuous at\[x=1\] done clear
  C)
             \[\underset{x\to 1}{\mathop{\lim }}\,f(x)=1\]                            done clear
  D)
             None of these done clear
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• question_answer41)
  If \[f(x)=\left\{ \begin{align}   & \frac{{{x}^{2}}+3x-10}{{{x}^{2}}+2x-15},\ \ \text{when }x\ne -5 \\  & \,\,a\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\text{when }x=-5 \\ \end{align} \right.\]is continuous at \[x=-5\], then the value of 'a' will be         [MP PET 1987]
  A)
             \[\frac{3}{2}\] done clear
  B)
             \[\frac{7}{8}\] done clear
  C)
             \[\frac{8}{7}\] done clear
  D)
             \[\frac{2}{3}\] done clear
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• question_answer42)
  If \[f(x)=\left\{ \begin{align}   & x+\lambda ,\ x\,<3 \\  & \,\,\,\,\,\,\,\,\,4,\,\,x=3 \\  & 3x-5,\,\,x>3 \\ \end{align} \right.\]is continuous at\[x=3\], then \[\lambda =\] [MP PET 1994, 2001; RPET 1999]
  A)
             4 done clear
  B)
             3 done clear
  C)
             2 done clear
  D)
             1 done clear
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• question_answer43)
  The value of k which makes \[f(x)=\left\{ \begin{align}   & \sin \frac{1}{x},\ x\ne 0 \\  & \,\,\,\,\,\,\,\,k,\,x=0 \\ \end{align} \right.\] continuous at \[x=0\]is                                                                         [MNR 1995]
  A)
             8 done clear
  B)
             1 done clear
  C)
             ?1 done clear
  D)
             None of these done clear
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• question_answer44)
  If \[f(x)=\left\{ \begin{align}   & \sin x,\ x\ne n\pi ,\ \ n\in Z \\  & \,\,\,\,\,\,2,\,\text{otherwise} \\ \end{align} \right.\] and \[g(x)=\left\{ \begin{align}   & {{x}^{2}}+1,\ x\ne 0,\,2 \\  & \,\,\,\,\,\,\,\,\,4,\,x=0 \\  & \,\,\,\,\,\,\,\,\,\,5,x=2 \\ \end{align} \right.,\]  then \[\underset{x\to 0}{\mathop{\lim }}\,g\,\{f(x)\}\] is [Kurukshetra CEE 1996]
  A)
             5 done clear
  B)
             6 done clear
  C)
             7 done clear
  D)
             1 done clear
  View Solution play_arrow

• question_answer45)
  Let \[f(x)=\left\{ \begin{align}   & \frac{x-4}{|x-4|}+a,\ x<4 \\  & \,\,\,\,\,\,\,\,\,\,\,\,a+b,\,x=4 \\  & \frac{x-4}{|x-4|}+b,\,x>4 \\ \end{align} \right.\]. Then \[f(x)\]is continuous at \[x=4\] when
  A)
             \[a=0,\ b=0\] done clear
  B)
             \[a=1,\ b=1\] done clear
  C)
             \[a=-1,\ b=1\] done clear
  D)
             \[a=1,\ b=-1\] done clear
  View Solution play_arrow

• question_answer46)
  Let \[f(x)=\left\{ \begin{align}   & \frac{{{x}^{4}}-5{{x}^{2}}+4}{|(x-1)(x-2)|},\ \ x\ne 1,\ 2 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,6,\,\,\,x=1 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,12,\,\,\,x=2 \\ \end{align} \right.\] Then \[f(x)\]is continuous on the set
  A)
             R done clear
  B)
             \[R-\{1\}\] done clear
  C)
             \[R-\{2\}\] done clear
  D)
             \[f:R\to R\] done clear
  View Solution play_arrow

• question_answer47)
  The value of \[f(0)\], so that the function \[f(x)=\frac{{{(27-2x)}^{1/3}}-3}{9-3{{(243+5x)}^{1/5}}},\,(x\ne 0)\]is continuous, is given by
  A)
             \[2/3\] done clear
  B)
             6 done clear
  C)
             2 done clear
  D)
             4 done clear
  View Solution play_arrow

• question_answer48)
  If the function \[f(x)=\left\{ \begin{align}   & {{(\cos x)}^{1/x}},\ x\ne 0 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,k,\,x=0 \\ \end{align} \right.\]is continuous at \[x=0\], then the value of k is                     [Kurukshetra CEE 1996]
  A)
             1 done clear
  B)
             ?1 done clear
  C)
             0 done clear
  D)
             e done clear
  View Solution play_arrow

• question_answer49)
  Function \[f(x)=\left\{ \begin{align}   & \,\,\,x-1,\ x<2 \\  & 2x-3,\,x\ge 2 \\ \end{align} \right.\]is  a continuous function [MP PET 1996]
  A)
             For all real values of x               done clear
  B)
             For \[x=2\]only done clear
  C)
             For all real values of x such that \[x\ne 2\] done clear
  D)
             For all integral values of x only done clear
  View Solution play_arrow

• question_answer50)
  If the function \[f(x)=\left\{ \begin{align}   & 1+\sin \frac{\pi x}{2}\,\,,\,\text{for}\,-\infty <x\le 1 \\  & \,\,\,\,\,\,\,\,ax+b,\,\text{for}\,1<x<3 \\  & \,\,\,\,6\tan \frac{x\pi }{12},\,\text{for}3\le x<6 \\ \end{align} \right.\] is continuous in the interval \[(-\infty ,\,6)\], then the values of a and b are respectively [MP PET 1998]
  A)
             0, 2 done clear
  B)
             1, 1 done clear
  C)
             2, 0 done clear
  D)
             2, 1 done clear
  View Solution play_arrow

• question_answer51)
  If \[f(x)=\left\{ \begin{align}   & x\sin \frac{1}{x},\,\,x\ne 0 \\  & \,\,\,\,\,\,\,\,\,\,\,\,k,\,\,x=0 \\ \end{align} \right.\]is continuous at \[x=0\], then the value of k is [MP PET 1999; AMU 1999; RPET 2003]
  A)
             1 done clear
  B)
             ?1 done clear
  C)
             0 done clear
  D)
             2 done clear
  View Solution play_arrow

• question_answer52)
  If \[f(x)=\left\{ \begin{align}   & \,\,\,\,\,\,\,\frac{\sin [x]}{[x]+1},\,\,\text{for}\,x>0 \\  & \frac{\cos \frac{\pi }{2}[x]}{[x]},\,\,\text{for}\,x<0 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,k,\,\,\text{at}\,x=0 \\ \end{align} \right.\]; where [x] denotes the greatest integer less than or equal to x, then in order that f be continuous at \[x=0\], the value of k is [Kurukshetra CEE 1998]
  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_answer53)
  The function \[f(x)\,=\left\{ \begin{align}   & x+2\,\,\,\,,\,\,\,1\le x\le 2 \\  & 4\,\,\,\,\,\,\,\,\,\,\,,\,\,\,x=2 \\  & 3x-2\,\,,\,\,\,x>2 \\ \end{align} \right.\] is continuous at [DCE 1999]
  A)
             \[x=2\] only done clear
  B)
             \[x\le 2\] done clear
  C)
             \[x\ge 2\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer54)
  If the function \[f(x)=\,\left\{ \begin{matrix}    5x-4 & , & \text{if} & 0<x\le 1  \\    4{{x}^{2}}+3bx & , & \text{if} & 1<x<2  \\ \end{matrix} \right.\] is continuous at every point of its domain, then the value of b is                    [RPET 2000]
  A)
             ? 1 done clear
  B)
             0 done clear
  C)
             1 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer55)
  The values of A and B such that the function \[f(x)=\left\{ \begin{matrix}    -2\sin x, & x\le -\frac{\pi }{2}  \\    A\sin x+B, & -\frac{\pi }{2}<x<\frac{\pi }{2}  \\    \cos x, & x\ge \frac{\pi }{2}  \\ \end{matrix} \right.\], is continuous everywhere are                                           [Pb. CET 2000]
  A)
             \[A=0,\,B=1\] done clear
  B)
             \[A=1,\,B=1\] done clear
  C)
             \[A=-1,\,B=1\] done clear
  D)
             \[A=-1,\,B=0\] done clear
  View Solution play_arrow

• question_answer56)
  If \[f(x)=\frac{{{x}^{2}}-10x+25}{{{x}^{2}}-7x+10}\] for \[x\]\[\ne \,\]5 and f is continuous at \[x=5,\] then \[f(5)=\] [EAMCET 2001]
  A)
             0 done clear
  B)
             5 done clear
  C)
             10 done clear
  D)
             25 done clear
  View Solution play_arrow

• question_answer57)
  In order that the function \[f(x)={{(x+1)}^{\cot \,x}}\] is continuous at \[x=0\], \[f(0)\] must be defined as [UPSEAT 2000; Kurukshetra CEE 2001; Pb. CET 2004]
  A)
             \[f(0)=\frac{1}{e}\] done clear
  B)
             \[f(0)=0\]            done clear
  C)
             \[f(0)=e\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer58)
  The function \[f(x)=\sin |x|\] is                  [DCE 2002]
  A)
             Continuous for all x done clear
  B)
             Continuous only at certain points done clear
  C)
             Differentiable at all points done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer59)
  If \[f(x)=\,|x|\], then \[f(x)\] is [DCE 2002]
  A)
             Continuous for all x done clear
  B)
             Differentiable at \[x=0\] done clear
  C)
                     Neither continuous nor differentiable at \[x=0\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer60)
  If \[f(x)=\left\{ \begin{matrix}    \frac{1-\sin x}{\pi -2x}, & x\ne \frac{\pi }{2}  \\    \,\,\,\,\,\,\,\,\,\,\,\,\,\lambda \,, & x=\frac{\pi }{2}  \\ \end{matrix} \right.\], be continuous at \[x=\pi /2,\] then value of \[\lambda \] is                                           [RPET 2002]
  A)
             ?1 done clear
  B)
             1            done clear
  C)
             0 done clear
  D)
             2 done clear
  View Solution play_arrow

• question_answer61)
  Let \[f(x)=\,\left\{ \begin{matrix}    \frac{\sin \pi x}{5x}, & x\ne 0  \\    k, & x=0  \\ \end{matrix} \right.\]. If \[f(x)\] is continuous at \[x=0,\] then \[k=\]  [Karnataka CET 2002]
  A)
             \[\frac{\pi }{5}\] done clear
  B)
             \[\frac{5}{\pi }\] done clear
  C)
             1 done clear
  D)
             0 done clear
  View Solution play_arrow

• question_answer62)
  If \[f(x)\,=\frac{2-\sqrt{x+4}}{\sin 2x},\,\,(x\ne 0),\] is continuous function at \[x=0\], then \[f(0)\] equals    [MP PET 2002]
  A)
             \[\frac{1}{4}\] done clear
  B)
             \[-\frac{1}{4}\] done clear
  C)
             \[\frac{1}{8}\] done clear
  D)
             \[-\frac{1}{8}\] done clear
  View Solution play_arrow

• question_answer63)
  If function \[f(x)=\left\{ \begin{matrix}    x\,\,\,\,\,, & \text{if}\,x\,\text{is rational}  \\    1-x, & \text{if}\,x\,\text{is irrational}  \\ \end{matrix}, \right.\] then \[f(x)\] is continuous at ...... number of points [UPSEAT 2002]
  A)
             \[\infty \] done clear
  B)
             1 done clear
  C)
             0 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer64)
  If \[f(x)=\left\{ \begin{matrix}    \frac{{{x}^{2}}-9}{x-3}\,, & \text{if }x\ne 3  \\    2x+k\,, & \text{otherwise}  \\ \end{matrix} \right.\], is continuous at \[x=3,\] then \[k=\]                  [Kerala (Engg.) 2002]
  A)
             3 done clear
  B)
             0 done clear
  C)
             ?6 done clear
  D)
             1/6 done clear
  View Solution play_arrow

• question_answer65)
  The function defined by \[f(x)\,=\,\left\{ \begin{matrix}    {{\left( {{x}^{2}}+{{e}^{\frac{1}{2-x}}} \right)}^{-1}} & , & x\ne 2  \\    k & , & x=2  \\ \end{matrix} \right.\], is continuous from right at the point x = 2, then k is equal to [Orissa JEE 2002]
  A)
             0 done clear
  B)
             1/4 done clear
  C)
             ?1/4 done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer66)
  For the function \[f(x)\,=\frac{{{\log }_{e}}(1+x)-{{\log }_{e}}(1-x)}{x}\] to be continuous at \[x=0,\] the value of \[f(0),\] should be  [MP PET 2003]
  A)
             ?1 done clear
  B)
             0 done clear
  C)
             ?2 done clear
  D)
             2 done clear
  View Solution play_arrow

• question_answer67)
  If \[f(x)\,=\,\left\{ \begin{matrix}    \frac{\sqrt{1+kx}-\sqrt{1-kx}}{x} & \text{,for}-1\le x<0  \\    2{{x}^{2}}+3x-2 & \text{,}\,\text{for }\,0\le \,x\le 1  \\ \end{matrix} \right.\], is continuous at \[x=0\],  then \[k=\] [EAMCET 2003]
  A)
             ? 4 done clear
  B)
             ? 3 done clear
  C)
             ? 2 done clear
  D)
             ? 1 done clear
  View Solution play_arrow

• question_answer68)
  The function \[f(x)=\frac{1-\sin x+\cos x}{1+\sin x+\cos x}\] is not defined at \[x=\pi .\] The value of \[f(\pi ),\] so that \[f(x)\] is continuous at \[x=\pi \], is [Orissa JEE 2003]
  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_answer69)
  If \[f(x)=\left\{ \begin{align}   & \frac{1-\cos x}{x},\,x\ne 0 \\  & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,k,\,x=0 \\ \end{align} \right.\]is continuous at \[x=0\] then \[k=\] [Karnataka CET 2004]
  A)
             0 done clear
  B)
             \[\frac{1}{2}\] done clear
  C)
             \[\frac{1}{4}\] done clear
  D)
             \[-\frac{1}{2}\] done clear
  View Solution play_arrow

• question_answer70)
  A function f on R into itself is continuous at a point a in R, iff for each \[\in >0\], there exists, \[\delta >0\]such that [UPSEAT 2004]
  A)
             \[|f(x)-f(a)|<\in \]Þ\[|x-a|<\delta \] done clear
  B)
             \[|f(x)-f(a)|>\in \]Þ\[|x-a|>\delta \] done clear
  C)
             \[|x-a|>\delta \]Þ\[|f(x)-f(a)|>\in \] done clear
  D)
             \[|x-a|<\delta \]Þ\[|f(x)-f(a)|<\in \] done clear
  View Solution play_arrow

• question_answer71)
  For the function \[f(x)=\left\{ \begin{align}   & \frac{{{e}^{1/x}}-1}{{{e}^{1/x}}+1},\,\,x\ne 0 \\  & 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,\,\,x=0 \\ \end{align} \right.\], which of the following is correct [MP PET 2004]
  A)
             \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\]does not exist          done clear
  B)
             \[f(x)\]is continuous at \[x=0\] done clear
  C)
             \[\underset{x\to 0}{\mathop{\lim }}\,f(x)=1\]                            done clear
  D)
             \[\underset{x\to 0}{\mathop{\lim }}\,f(x)\]exists but \[f(x)\]is not continuous at \[x=0\] done clear
  View Solution play_arrow

• question_answer72)
  The function 'f' is defined by \[f(x)=2x-1,\]if \[x>2\], \[f(x)=k\ \]if\[x=2\]and \[{{x}^{2}}-1,\]if \[x<2\]is continuous, then the value of k is equal to [Pb. CET 2002]
  A)
             2 done clear
  B)
             3 done clear
  C)
             4 done clear
  D)
             ?3 done clear
  View Solution play_arrow

• question_answer73)
  In the function \[f(x)=\frac{2x-{{\sin }^{-1}}x}{2x+{{\tan }^{-1}}x},\ (x\ne 0)\]is continuous at each point of its domain, then the value of \[f(0)\] is [RPET 2000]
  A)
             2 done clear
  B)
             \[1/3\] done clear
  C)
             \[2/3\] done clear
  D)
             \[-1/3\] done clear
  View Solution play_arrow

• question_answer74)
  The function \[f(x)=|x|+\frac{|x|}{x}\] is [Karnataka CET 2003]
  A)
             Continuous at the origin done clear
  B)
             Discontinuous at the origin because |x| is discontinuous there done clear
  C)
             Discontinuous at the origin because \[\frac{|x|}{x}\] is discontinuous there done clear
  D)
             Discontinuous at the origin because both |x| and \[\frac{|x|}{x}\] are discontinuous there done clear
  View Solution play_arrow

• question_answer75)
  The value of f at \[x=0\] so that the function \[f(x)=\frac{{{2}^{x}}-{{2}^{-x}}}{x},x\ne 0\], is continuous at \[x=0\], is [Kerala (Engg.) 2005]
  A)
             0 done clear
  B)
             log 2 done clear
  C)
             4 done clear
  D)
             \[{{e}^{4}}\] done clear
  E)
             log 4 done clear
  View Solution play_arrow

• question_answer76)
  The function \[f(x)=\frac{2{{x}^{2}}+7}{{{x}^{3}}+3{{x}^{2}}-x-3}\]is discontinuous for [J&K 2005]
  A)
             \[x=1\] only                                done clear
  B)
             \[x=1\] and \[x=-1\] only done clear
  C)
             \[x=1,x=-1,x=-3\] only done clear
  D)
             \[x=1,x=-1,x=-3\] and other values of x done clear
  View Solution play_arrow

• question_answer77)
  Let \[f(x)=\left\{ \begin{align}   & {{x}^{p}}\sin \frac{1}{x},x\ne 0 \\  & 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,,x=0 \\ \end{align} \right.\] then \[f(x)\]is continuous but not differential at \[x=0\] if [DCE 2005]
  A)
             \[0<p\le 1\] done clear
  B)
             \[1\le p<\infty \] done clear
  C)
             \[-\infty <p<0\] done clear
  D)
             p = 0 done clear
  View Solution play_arrow

• question_answer78)
  If \[f(x)=\]\[\left\{ \begin{align}   & \frac{1-(x)}{1+x},\,\,\,\,\,x\ne -1 \\  & \,\,1\,\,\,\,\,\,\,\,\,,\,\,\,\,\,x=-1 \\ \end{align} \right.\], then  the value of \[f(|2k|)\]will be (where [ ] shows the greatest integer function) [DCE 2005]
  A)
             Continuous at \[x=-1\] done clear
  B)
             Continuous at \[x=0\] done clear
  C)
             Discontinuous at \[x=\frac{1}{2}\] done clear
  D)
             All of these done clear
  View Solution play_arrow

• question_answer79)
  Function \[f(x)=\frac{1-\cos 4x}{8{{x}^{2}}},\] where \[x\ne 0\]and \[f(x)=k\] where \[x=0\] is a continous function at \[x=0\]then the value of k will be [AMU 2005]
  A)
             \[k=0\] done clear
  B)
             \[k=1\] done clear
  C)
             \[k=-1\] done clear
  D)
             None of these done clear
  View Solution play_arrow

• question_answer80)
  If \[f(x)=\left\{ \begin{align}   & \,\,\,\,\,\,\,{{e}^{x}};\,\,\,\,x\le 0 \\  & |1-x|;\,\,x>0 \\ \end{align} \right.\], then [Roorkee 1995]
  A)
             \[f(x)\] is differentiable at \[x=0\] done clear
  B)
             \[f(x)\] is continuous at \[x=0\] done clear
  C)
             \[f(x)\] is differentiable at \[x=1\] done clear
  D)
                  \[f(x)\] is continuous at \[x=1\] done clear
  View Solution play_arrow