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question_answer1)
Which of the following functions have finite number of points of discontinuity in R ([.] represents the greatest integer function)?
A)
\[\operatorname{tanx}\] done
clear
B)
\[x[x]\] done
clear
C)
\[\frac{\left| x \right|}{x}\] done
clear
D)
\[\sin [\pi x]\] done
clear
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question_answer2)
\[f(x)=\underset{n\to \infty }{\mathop{\lim }}\,\frac{{{(x-1)}^{2n}}-1}{{{(x-1)}^{2n}}+1}\]is discontinuous at
A)
\[x=0\]only done
clear
B)
\[x=2\]only done
clear
C)
\[x=0\]and 2 done
clear
D)
none of these done
clear
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question_answer3)
The value of f(0) so that the function \[f(x)=\frac{2x-{{\sin }^{-1}}x}{2x+{{\tan }^{-1}}x}\]is continuous at each point in its domain, is equal to
A)
2 done
clear
B)
1/3 done
clear
C)
2/3 done
clear
D)
-1/3 done
clear
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question_answer4)
The function \[f(x)=\frac{{{({{3}^{x}}-1)}^{2}}}{\sin x\cdot \ln \,(1+x)},x\ne 0\], is continuous at x=0. Then the value of f(0) is
A)
\[2{{\log }_{e}}3\] done
clear
B)
\[{{(2{{\log }_{e}}3)}^{2}}\] done
clear
C)
\[{{\log }_{e}}6\] done
clear
D)
none of these done
clear
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question_answer5)
The function f(x)=|2 sgn 2x|+2 has
A)
jump discontinuity done
clear
B)
removal discontinuity done
clear
C)
infinite discontinuity done
clear
D)
no discontinuity at x=0 done
clear
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question_answer6)
Let \[f(x)=\left\{ \begin{matrix} \frac{x-4}{\left| x-4 \right|}+a,\,\,\,\,\,\,x4 \\ \end{matrix} \right.\] Then \[f(x)\] is continuous at x=4
A)
\[a=0,\text{ }b=0\] done
clear
B)
\[a=1,\text{ }b=1\] done
clear
C)
\[a=-\,1,\text{ }b=1\] done
clear
D)
\[a=1,\text{ }b=-1\] done
clear
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question_answer7)
A point where function \[f(x)=[sin[x]]\] is not continuous in \[(0,2\pi )\], [.] denotes the greatest integer \[\le x\], is
A)
(3, 0) done
clear
B)
(2, 0) done
clear
C)
(1, 0) done
clear
D)
none of these done
clear
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question_answer8)
If \[f(x)=\left\{ \begin{matrix} x+2,\,\,\,\,\,\,\,\,\,x<0 \\ -{{x}^{2}}-2,\,\,\,\,\,\,0\le x<1 \\ x,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\ge 1 \\ \end{matrix} \right.\], Then the number, of points of discontinuity of \[\left| f(x) \right|\]is
A)
1 done
clear
B)
2 done
clear
C)
3 done
clear
D)
none of these done
clear
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question_answer9)
The function \[f(x)\] defined by\[f(x)=\left\{ \begin{matrix} {{\log }_{(4x-3)}}({{x}^{2}}-2x+5),\,\,\,\,\,\,\frac{3}{4}<x<1\,\,\text{and}\,\,x>1 \\ 4,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x=1 \\ \end{matrix} \right.\]
A)
is continuous at x=1 done
clear
B)
is discontinuous at x=1 since \[f({{1}^{+}})\]does not exist though \[f({{1}^{-}})\]exists done
clear
C)
is discontinuous at x=1 since \[f({{1}^{-}})\]does not exist thought \[f({{1}^{+}})\]exists done
clear
D)
is discontinuous at x=1 since neither \[f({{1}^{+}})\]nor \[f({{1}^{-}})\]exists. done
clear
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question_answer10)
If \[f(x)=\frac{{{x}^{2}}-bx+25}{{{x}^{2}}-7x+10}\] for \[x\ne 5\] is continuous at x=5, then the value of \[f(5)\] is
A)
0 done
clear
B)
5 done
clear
C)
10 done
clear
D)
25 done
clear
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question_answer11)
If \[f(x)={{x}^{3}}\sgn x,\]then
A)
f is derivable at x=0 done
clear
B)
f is continuous but not derivable at x=0 done
clear
C)
LHD at x=0 is 1 done
clear
D)
RHD at x=0 is 1 done
clear
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question_answer12)
The number of values of \[x\in [0,2]\] at which \[f(x)=\left| x-\frac{1}{2} \right|+\left| x-1 \right|+\tan x\] is not differentiable is
A)
0 done
clear
B)
1 done
clear
C)
3 done
clear
D)
none of these done
clear
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question_answer13)
\[f(x)=\left\{ \begin{matrix} \frac{x}{2{{x}^{2}}+\left| x \right|,}\,\,x\ne 0 \\ 1.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x=0 \\ \end{matrix} \right.\]. Then \[f(x)\] is
A)
continuous but non-differentiable at x=0 done
clear
B)
differentiable at x=0 done
clear
C)
discontinuous at x=0 done
clear
D)
none of these done
clear
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question_answer14)
Which of the following function is not differentiable at x=1?
A)
\[f(x)=({{x}^{2}}-1)\left| (x-1)(x-2) \right|\] done
clear
B)
\[f(x)=sin(\left| x-1 \right|)-\left| x-1 \right|\] done
clear
C)
\[f(x)=\tan (\left| x-1 \right|)-\left| x-1 \right|\] done
clear
D)
None of these done
clear
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question_answer15)
If \[f(x)=\left\{ \begin{matrix} {{x}^{3}},\,\,{{x}^{2}}<1 \\ x,\,\,\,{{x}^{2}}>1 \\ \end{matrix} \right.\], then \[f(x)\] is differentiable at
A)
\[(-\infty ,\infty )-\{1\}\] done
clear
B)
\[(-\infty ,\infty )\tilde{\ }\{1,-1\}\] done
clear
C)
\[(-\infty ,\infty )\tilde{\ }\{1,-1,0\}\] done
clear
D)
\[(-\infty ,\infty )\tilde{\ }\{-1\}\] done
clear
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question_answer16)
If \[f(x)=\left\{ \begin{matrix} {{e}^{{{x}^{2}}+x}}\,\,\,\,\,x>0 \\ ax+b,\,\,x\le 0 \\ \end{matrix} \right.\]is differentiable at x=0, then
A)
\[a=1,\text{ }b=-\,1\] done
clear
B)
\[a=-1,\text{ }b=1\] done
clear
C)
\[a=1,\text{ }b=1\] done
clear
D)
\[a=-\,1,\text{ }b=-\,1\] done
clear
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question_answer17)
If \[f(x)=\left\{ \begin{matrix} x-1,\,\,\,\,\,\,\,x<0 \\ {{x}^{2}}-2x,\,\,x\ge 0 \\ \end{matrix} \right.,\], then
A)
\[f(\left| x \right|)\]is discontinuous at x=0 done
clear
B)
\[f(\left| x \right|)\]is differentiable at x=0 done
clear
C)
\[|f(x)|\]is non-differentiable at x=0, 2 done
clear
D)
\[|f(x)|\]is continuous at x=0 done
clear
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question_answer18)
Let \[f(x)=\underset{n\to \infty }{\mathop{\lim }}\,\frac{\log (2+x)-{{x}^{2n}}\sin x}{1+{{x}^{2n}}}\], then
A)
f is continuous at x=1 done
clear
B)
\[\underset{x\to {{1}^{+}}}{\mathop{\lim }}\,f(x)=log3\] done
clear
C)
\[\underset{x\to {{1}^{+}}}{\mathop{\lim }}\,f(x)=-\sin 1\] done
clear
D)
\[\underset{x\to {{1}^{-}}}{\mathop{\lim }}\,f(x)\]does not exist done
clear
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question_answer19)
If \[f(x)=\left\{ \begin{matrix} {{x}^{2}}-ax+3,\,\,\,x\,\,\text{is}\,\,\text{rational} \\ 2-x,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\,\,\text{is}\,\,\text{irrational} \\ \end{matrix} \right.\]is continuous at exactly two points, then the possible values of a are
A)
\[(2,\infty )\] done
clear
B)
\[(-\infty ,3)\] done
clear
C)
\[(-\infty ,-3)\cup (3,\infty )\] done
clear
D)
none of these done
clear
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question_answer20)
The set of all points where \[f(x)=\sqrt[3]{{{x}^{2}}\left| x \right|}-\left| x \right|-1\]is not differentiable is
A)
{0} done
clear
B)
\[\left\{ -1,\text{ }0,\text{ }1 \right\}\] done
clear
C)
{0, 1} done
clear
D)
none of these done
clear
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question_answer21)
If the function \[f(x)=\frac{{{(128a+ax)}^{1/8}}-2}{{{(32+bx)}^{1/5}}-2}\] is continuous at x=0, then the value of \[\frac{a}{b}=kf(0)\]. The value of k is____.
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question_answer22)
Let \[f(x)={{x}^{3}}-{{x}^{2}}-3x-1,\,\,g(x)=(x+1)\,a\] and \[h(x)=\frac{f(x)}{g(x)}\]where h is a rational function such that: |
(i) it is continuous every where except when \[x=-1\,,\] |
(ii) \[\underset{x\to \infty }{\mathop{\lim }}\,h(x)=\infty \] and |
(iii) \[\underset{x\to \infty }{\mathop{\lim }}\,h(x)=\frac{1}{2}\]. |
The value of h(1) is____. |
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question_answer23)
Total number of points of non-differentiability of \[f(x)=\min .\{1,\,\,1+{{x}^{3}},{{x}^{2}}-3x+3\}\]is ______.
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question_answer24)
Suppose \[f(x)\] is differentiable at x=1 and\[\underset{h\to 0}{\mathop{\lim }}\,\frac{1}{h}f(1+h)=5\], then f?(1) equals _______.
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question_answer25)
The function f:R-{0}\[\to \]R given by \[f(x)=\frac{1}{x}-\frac{2}{{{e}^{2x}}-1}\] can be made continuous at x=0 by defining f(0) as _______.
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