Question 1 :
If $f(x)=\displaystyle\frac{\log(1+ax)-log(1-bx)}{x}$ for $x\neq 0$ and $f(0)=k$ and $f(x)$ is continuous at $x=0$, then $k$ is equal to:<br>
Question 2 :
Function $f(x) = \left\{\begin{matrix}(\log_{2}2x)^{\log_{x} 8};& x\neq 1\\(k - 1)^{3};& x = 1\end{matrix}\right.$ is continuous at $x = 1$, then $k =$ _______.<br>
Question 3 :
If $f(x)=\displaystyle \frac{\sin x}{x},x\neq 0$ is to be continuous at ${x}=0$ then $\mathrm{f}({0})=$<br/>
Question 4 :
The integer $'n'$ for which $\mathop {\lim }\limits_{x \to 0} \dfrac{{\cos 2x - 1}}{{{x^n}}}$ is a finite non-zero number is
Question 5 :
If $f(x) = (1 + 2x)^{\frac{1}{x}}$, for $x \neq 0$ is continuous at $x = 0$, then $f(0) = $_______.
Question 6 :
$f(x)=\left\{\begin{matrix} 2x-1& if &x>2 \\ k & if &x=2 \\  x^{2}-1& if & x<2\end{matrix}\right.$is continuous at $x= 2$ then $k =$
Question 7 :
If as continuous function 'f' satisfies the realation <br/>$\int_{0}^{t}(f(x) \, - \, \sqrt{f^1(x)})dx \, = \, 0 \, and \, f(0) \, = \, {-1}$<br/>the f(x) is equal to
Question 8 :
If $f(x) = \begin{cases}\dfrac{x^2-(a+2)x+a}{x-2} & x\ne 2\\ 2 & x = 2 \end{cases}$ is continuous at $x = 2$, then the value of $a$ is
Question 9 :
If $f\left( x \right) =\dfrac { 3sinx-sin\left( 3x \right) }{ { 2x }^{ 3 } } ,x\neq 0,f\left( 0 \right) =2,$ at $x=0,f$ is
Question 10 :
The function $f\left( x \right)=\left[ x \right] ,$  at ${ x }=5$ is:<br/>
Question 11 :
Following function is continous at the point $x=2$ <br/>          $f\left( x \right) = \left\{ \begin{array}{l}1 + x,\,\,\,\,when\,\,x < 2\\5 - x,\,\,\,\,when\,\,x \ge 2\end{array} \right.$
Question 12 :
If $ f\left( x \right) =\begin{cases} 1 & x<0 \\ { x }^{ 2 } & x\ge 0 \end{cases}$ then at $x=0$
Question 13 :
If the function $\displaystyle f(x)=\begin{cases} 2x^{2}-3 & \text{ if } 0<x\leq 1 \\ x^{2}+bx-1 & \text{ if } 1<x<2 \end{cases}$ is continuous at every point of its domain then the value of $b$ is 
Question 14 :
The function $f(x)=\dfrac{1-\sin x+\cos x}{1+\sin x+\cos x}$ is not defined at $x=\pi$. The value of $f(\pi)$ so that $f(x)$ is continuous at $x=\pi$ is
Question 16 :
<br/>The function $f(x)=\begin{cases}(1+x)^{\frac{5}{x}}& for \ {x}\neq 0,\\ e^{5} & for \  {x}=0\end{cases}$ is ........ at ${x}=0$<br/>
Question 17 :
If $f(x)=\begin{cases} \sin { x } \quad if\quad x\le 0 \\ { x }^{ 2 }+{ a }^{ 2 }\quad if\quad 0<x<1 \\ bx+2\quad if\quad 1\le x\le 2 \\ 0\quad if\quad x>2 \end{cases}$ is continuous on $R$, then $a+b+ab=$
Question 18 :
Let $[x]$ denotes the greatest integer less than or equal to $x$. Then, the value of $\alpha$ for which the function $f(x)=\begin{cases}\dfrac{\sin[-x^2]}{[-x^2]} &x\neq 0 \\ \alpha&x=0 \end{cases}$ is continuous at $x = 0$, is
Question 19 :
Let $f: R \rightarrow R$ be defined by $f(x) = \left\{\begin{matrix}\alpha + \dfrac {\sin [x]}{x}&if\ x > 0\\ 2& if\ x = 0\\ \beta + \left [\dfrac {\sin x - x}{x^{3}}\right ]& if\ x < 0\end{matrix}\right.$<br>where $[x]$ denotes the integral part of y. If f is continuous at $x = 0$, then $\beta - \alpha =$
Question 20 :
The value of $f$ at $x = 0$ so that function $f(x) = \dfrac {2^{x} - 2^{-x}}{x}, x \neq 0$, is continuous at $x = 0$, is
Question 21 :
If $f(x)=\begin{cases} x+2,\quad \quad when\quad x<1 \\ 4x-1,\quad when\quad 1\le x\le 3 \\ { x }^{ 2 }+5,\quad when\quad x>3 \end{cases}$, then correct statement is-
Question 22 :
If $f : [-2, 2]\rightarrow R$ is defined by $f(x) =\begin{cases} \dfrac { \sqrt { 1+ex }-\sqrt{1-ex} }{x} &for& -2\le x < 0 \\ \dfrac{x+3}{ x+1}&for& 0 \le x \le 2\end{cases}$<br/>is continuous on$ [-2,2]$, then $e=$
Question 23 :
If $f(x)=\begin{cases} mx-1,\quad x\le 5 \\ 3x-5,\quad x>5 \end{cases} $ is continuous then value of m is:
Question 24 :
If $f(x) = \dfrac{x \, \sin 5 x}{\tan \, 2x \, \tan 7 x} , x \neq 0 , \, f(0) = \dfrac{5}{9} $ then at x = 0 f(x) is
Question 25 :
If the function $\displaystyle f\left( x \right)=\begin{cases} \begin{matrix} 1+\sin { \dfrac { \pi }{ 2 } x } , for & -\infty <x\le 1 \end{matrix} \\ \begin{matrix} ax+b, for & 1<x<3 \end{matrix} \\ \begin{matrix} 6\tan { \dfrac { \pi x }{ 12 } } ,for & 3\le x<6 \end{matrix} \end{cases}$ is continous in the interval $(-\infty,6)$ then the value of $a$ and $b$ are?
Question 26 :
Let $f(x)=\begin{cases} -2\sin { x } ,\quad \quad if\quad x\le -\cfrac { \pi }{ 2 } \\ A\sin { x } +B,\quad if\quad -\cfrac { \pi }{ 2 } <x<\cfrac { \pi }{ 2 } \\ \cos { x } ,\quad if\quad x\ge \cfrac { \pi }{ 2 } \end{cases}$. Then
Question 27 :
Assertion: The function $y=f(x)$, defined parametrically as $y=t^2+t|t|,x=2t-|t|,t\in R,$ is continuous for all real $x.$
Reason: $f\left( x \right)=\begin{cases} \begin{matrix} 2{ x }^{ 2 }, & x\ge 0 \end{matrix} \\ \begin{matrix} 0, & x<0 \end{matrix} \end{cases}$
Question 28 :
The function $f\left( x \right)=\sin ^{ -1 }{ \left( \cos { x }  \right)  } $ is
Question 29 :
If $f(x)=\left\{\begin{matrix} \displaystyle\frac{1-\cos 4x}{x^2}, & when x < 0\\ a, & when x=0 \\ \displaystyle\frac{\sqrt{x}}{\sqrt{(16+\sqrt{x})}-4}, & when x> 0\end{matrix}\right.$ is continuous at $x=0$, then the value of a will be.<br>
Question 30 :
If $f(x) = \dfrac {x}{1 + x} + \dfrac {x}{(1 + x)(1 + 2x)} + \dfrac {x}{(1 + 2x)(1 + 3x)} + ....\infty$, then
Question 32 :
The function $f\left ( x \right )=\left | \sin x \right |\left ( -2\pi \leq x\leq 2\pi \right )$ is <br>
Question 33 :
Assertion: If $f\left( x \right)=sgn\left( x \right) $ and $g\left( x \right)=x\left( 1-{ x }^{ 2 } \right) ,$ then $fog\left( x \right) $ and $gof\left( x \right) $ are continuous everywhere 
Reason: $fog\left( x \right) =\begin{cases} \begin{matrix} -1, & x\in \left( -1,0 \right) \cup \left( 1,\infty  \right)  \end{matrix} \\ \begin{matrix} 0, &  \end{matrix}x\in \left\{ -1,0,1 \right\}  \\ \begin{matrix} 1, & x\in \left( -\infty ,-1 \right) \cup \left( 0,1 \right)  \end{matrix} \end{cases}$ and $gof\left( x \right) =0,\quad \forall x\in R$
Question 34 :
The values of $p$ and $q$ so that the function<br>$f(x)=\begin{cases} { \left( 1+\left| \sin { x } \right| \right) }^{ \cfrac { p }{ \sin { x } } },\cfrac { -\pi }{ 6 } <x<0 \\ q,\quad \quad \quad \quad x=0 \\ { e }^{ \cfrac { \sin { 2x } }{ \sin { 3x } } },\quad \quad 0<x<\cfrac { \pi }{ 6 } \end{cases}$ is continuous at $x=0$ is
Question 35 :
Find $m$ and $n$ it few$ = \left\{ \begin{array}{l}{x^2} + mx + n\,\,\,0 \le x \subset 2\\4x - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,2 \le x \subset 4\\4{x^2} + 17n\,\,\,\,\,\,4 \le x \subset 6\end{array} \right.$ is continous function
Question 37 :
Let$f\left ( x \right )=\left [ x \right ]+\left [ -x \right ]$.Then for any integer <b>n</b> andnon integer k
Question 38 :
If $f(x)=\left\{\begin{matrix} \dfrac{log_ex}{x-1} & x\neq 1\\ k & x=1\end{matrix}\right.$ continuous at $x=1$, then the value of k is?
Question 39 :
$f(x)=\dfrac{p+q^{\frac{1}{x}}}{r+s^{\frac{1}{x}}},<br>s<1, q<1,r\neq 0, \mathrm{f}(\mathrm{0})=1$, is left continuous at $x =0$ then<br>
Question 40 :
Let $f\left( x \right) = \mathop {\lim }\limits_{n \to \infty } {\left( {\sin x} \right)^{2n}}.$ Then, which one of  the following is incorrect ?
Question 41 :
If $f(x)=\begin{cases} a{ x }^{ 2 }-b\quad if\quad \left| x \right| <1 \\ -\cfrac { 1 }{ \left| x \right| } \quad if\quad \left| x \right| \ge 1\quad \end{cases}$ is derivable at $x=1$ then the values of $a+b$ is
Question 42 :
The value of k which makes $f(x)=\left\{\begin{matrix} \sin\dfrac{1}{x}, x\neq 0\\ k, x=0\end{matrix}\right.$ continuous at $x=0$ is?<br>
Question 43 :
The value of k that makes function f, defined below, continuous is $f(x) =\left\{\begin{matrix}\dfrac {2x^{2} + 5x}{x},&when\ x\neq 0\\ 3k - 1,& when\ x = 0\end{matrix}\right.$<br>
Question 44 :
If $\displaystyle f\left( x \right)=\begin{cases} \begin{matrix} \dfrac { x\log { \cos { x }  }  }{ \log { \left( 1+{ x }^{ 2 } \right)  }  } , & x\neq 0 \end{matrix} \\ \begin{matrix} 0, & x=0 \end{matrix} \end{cases}$ then 
Question 45 :
Evaluate : $\lim _ { x \rightarrow \infty } \left( 1 + \dfrac { \lambda } { x } + \dfrac { \mu } { x ^ { 2 } } \right) ^ { 2 x } = e ^ { 4 } \text { then } \lambda = ( \mu \in \mathbf { R } )$
Question 46 :
Let $f(x)=\begin{cases} \cfrac { { x }^{ 2 } }{ a }\>; \quad 0\le x<1 \\ a\>;\quad 1\le x<\sqrt { 2 } \\ \cfrac { 2{ b }^{ 2 }-4b }{ { x }^{ 2 } }\>;\quad \sqrt { 2 } \le x<\infty \end{cases}$<br>If $f(x)$ is continuous for $0\le x < \infty$, then the most suitable values of $a$ and $b$ are
Question 47 :
The function $f(x)=\begin{cases} \cfrac { { e }^{ 1/x }-1 }{ { e }^{ 1/x }+1 } \\ 0,\quad \quad x=0 \end{cases},\quad x\neq 0$
Question 48 :
Let $f\left( x \right) =\begin{cases} { x }^{ 3 }-{ x }^{ 2 }+10x-5\quad ,x\le 1 \\ -2x+\log _{ 2 }{ \left( { b }^{ 2 }-2 \right) ,x>1 }  \end{cases}$ the set of values of $b$ for which $f\left( x \right) $ has greatest <br/>value at $x=1$ is given by:
Question 49 :
If $f(x)$ is continuous in $[0, 1]$ and $\displaystyle f\left ( \dfrac{1}{3} \right )=1$ then $\displaystyle \lim_{n\to \infty }f\left ( \frac{n}{\sqrt{9n^{2}+1}} \right )$ is
Question 50 :
If $f(x)=\begin{cases} \cfrac { 1-\cos { 4x } }{ { x }^{ 2 } } \quad ,\quad \quad \quad x<0 \\ a\quad \quad \quad \quad \quad ,\quad \quad \quad x=0 \\ \cfrac { \sqrt { x } }{ \sqrt { 16+\sqrt { x } } -4 } \quad \quad \quad ,\quad x>0 \end{cases}$, then the correct statement is-
Question 51 :
The value of $k$ for which the function$\displaystyle f\left ( x \right )=\frac{\left ( e^{x} -1\right )^{4}}{\sin \left ( \frac{x^{2}}{k^{2}} \right )\log \left \{ 1+\left ( \frac{x^{2}}{2} \right ) \right \}},x\neq 0;f\left ( 0 \right )=8$may be continuous function is<br>
Question 52 :
Let $k$ be a non-zero real number. If $f(x) = \displaystyle \left\{\begin{matrix} \dfrac{(e^x - 1)^2}{\displaystyle \sin \left ( \frac{x}{k} \right ) \log \left ( 1 + \frac{x}{4} \right )}, & x \neq 0\\ 12, & x = 0\end{matrix}\right.$ is a continuous function, then the value of $k$ is
Question 53 :
Let $f\left( x \right)  = \dfrac { 1-\tan { x }  }{ 4x-\pi  } , x \neq  \pi /4and x \in \left[ 0, \pi /2 \right]  = \lambda , x= \pi /4$ if $f\left( x \right)$ is continuous in $\left( 0, \pi /2 \right) , then\lambda $ then
Question 54 :
Let $f\left ( x \right )=x-1$ and $g\left ( x \right )=\dfrac{1}{2}$. Then the set of points where $g(f(g(x)))$ is continuous is
Question 55 :
If $f(x)=\left\{\begin{matrix}mx +1, &x \leq \frac{\pi}{2} \\  sin \, x+n, & x >\frac{\pi}{2}\end{matrix}\right.$ is continuous at $x=\dfrac{\pi}{2}$, then<br/>
Question 56 :
Let $f(x)$ be a continuous function whose range is $[2, 6.5]$. If $\displaystyle h\left ( x \right )= \left [ \dfrac{\cos x+f\left ( x \right )}{\lambda } \right ], \lambda \in N $, be continuous, where $[ \cdot  ]$ denotes the greatest integer function, then the least value of $ \lambda$ is 
Question 57 :
$f(x)=\left\{\begin{matrix} -2\sin x& if & x\leq -\dfrac{\pi }{2}\\ a \sin x+b & if &-\dfrac{\pi }{2}<x<\dfrac{\pi }{2} \\  \cos x& if & x\displaystyle \geq\dfrac{\pi}{2}\end{matrix}\right.$ and $f(x)$ is continuous everywhere then $(a,b)$ =
Question 58 :
Let $f\left( x \right) =\begin{cases} { \left( x-1 \right) }^{ \dfrac { 1 }{ 2-x } },\quad x>1,\quad x\neq 2 \\ k\quad \quad \quad \quad \quad ,x=2 \end{cases}$<br>The value of $k$ for which $f$ is continuous at $x = 2$ is<br>
Question 59 :
$f(x)=\begin{cases} \left[ x \right] +\left[ -x \right] ,\quad when\quad \quad x\neq 2 \\ \lambda \quad \quad \quad \quad ,\quad when\quad \quad \quad \quad \quad \quad x=2 \end{cases}$<br>If $f(x)$ is continuous at $x=2$ then, the value of $\lambda$ will be
Question 60 :
The function of $f(x)=\left[ x \right] $ where $\left[ x \right] $ the greatest integer function is continuous at
Question 61 :
Let $\displaystyle f\left ( x \right )=\dfrac{1-\tan x}{4x-\pi }$, $x\neq \pi /4$, $\displaystyle x\in \left [ 0, \dfrac{\pi }{2} \right ]$.If $f(x)$ is continuous in $\displaystyle \left [ 0, \dfrac{\pi }{2} \right ]$ then $f\left ( \dfrac{\pi}{4} \right )$ is?<br>
Question 62 :
If $f : [-2, 2] \rightarrow R$ is defined by<br/>$f(x) = \left \{\begin{matrix} \dfrac{\sqrt{1 + cx} - \sqrt{1 - cx}}{x} &, & for & -2\le x < 0 \\ \dfrac{x + 3}{x + 1} &, & for & 0 \le x \le 2 \end{matrix} \right.$ is continuous on $[-2, 2]$ then $c =$
Question 63 :
Let $f\left( x \right)$ be a continuous function whose range is $\left[ 2,6,5 \right]$. If $h\left( x \right) =\left[ \dfrac { \cos { x } +f\left( x \right) }{ \lambda } \right]$, $\lambda \in N$ be continuous, where $\left[ . \right]$ denotes the greatest integer function, then the least value of $\lambda $ is
Question 64 :
The value of $ \displaystylef(0)$, so that function, $f(x)=\cfrac { \sqrt { { a }^{ 2 }-ax+{ x }^{ 2 } } -\sqrt { { a }^{ 2 }+ax+{ x }^{ 2 } } }{ \sqrt { a+x } -\sqrt { a-x } } $ becomes continuous for all $x$, is given by-
Question 66 :
<br>The function $\displaystyle \mathrm{f}({x})=\begin{cases}\dfrac{1-\sin x}{(\pi-2x)^{2}} & x \neq\dfrac{\pi}{2}\\ \mathrm{k}& {x}=\dfrac{\pi}{2}\end{cases}$ is continuous at $\displaystyle {x}=\dfrac{\pi}{2}$ then $\mathrm{k}$ is equal to<br>
