-
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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
View Solution play_arrow
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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