Question 1 :
$f(x)=\begin{cases} \cfrac { { x }^{ 5 }-32 }{ x-2 }  \\ k,\quad \quad x=2 \end{cases},\quad x\neq 2$ is continuous at $x=2$, then $k=.....$
Question 2 :
The function $f\left ( x \right )=\left | \sin x \right |\left ( -2\pi \leq x\leq 2\pi \right )$ is <br>
Question 3 :
If $f(x) =\begin{cases} \dfrac{\sin(p+1)x + \sin x}{x}& , &x < 0\\\quad \quad \quad q&,& x = 0\\ \dfrac{\sqrt{x^2 + x}- \sqrt{x}}{x^{3/2}}&,& x > 0\end{cases}$ is continuous at $x = 0$ the $(p, q)$ is
Question 4 :
The function  $\displaystyle f(x)=\begin{cases}\displaystyle \frac{x^{2}}{a} & 0\leq x< 1 \\ a & 1\leq x< \sqrt{2} \\ (2b^{2}-4b)/x^{2} & \sqrt{2}\leq x< \infty  \end{cases}$ is continuous for $0\leq x< \infty $, then the most suitable values of $a$ and $b$ are
Question 5 :
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 6 :
$f(x)=\dfrac{p+q^{\frac{1}{x}}}{r+s^{\frac{1}{x}}},<br>s<1, q<1,r\neq 0, \mathrm{f}(\mathrm{0})=1$, is left continuous at $x =0$ then<br>
Question 7 :
Let $f(x)=\cos2x.\cot\left (\displaystyle \frac{\pi }{4}-x \right )$ If $f$ is continuous at $x=\displaystyle \frac{\pi}{4}$ then the value of $f(\displaystyle \frac{\pi}{4})$is equal to
Question 8 :
Let $f(x)=\left\{\begin{matrix}<br>x, & if \, x \, is\, irrational\\ <br>0, & if\, is\, rational<br>\end{matrix}\right.$ then f is _____.<br>
Question 9 :
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 10 :
If the function $f(x)=\begin{cases} { \left( \cos { x } \right) }^{ 1/x } \\ k,\quad \quad x=0 \end{cases},\quad x\neq 0$ is continuous at $x=0$ then the value of $k$ is-
Question 11 :
If $\displaystyle f\left( x \right)=\begin{cases} \begin{matrix} x+\lambda , & -1<x<3 \end{matrix} \\ \begin{matrix} 4, & x=3 \end{matrix} \\ \begin{matrix} 3x-5, & x>3 \end{matrix} \end{cases}$ is continuous at $x=3$ then tha value of $\lambda$ is
Question 12 :
Consider the function $f(x) = \left\{\begin{matrix}\dfrac {\alpha \cos x}{\pi - 2x} & if & x\neq \dfrac {\pi}{2}\\ 3 & if & x = \dfrac {\pi}{2}\end{matrix}\right.$<br/>which is continuous at $x = \dfrac {\pi}{2}$, where $\alpha$ is a constant.What is the value of $\alpha$?
Question 13 :
Let $f(x)=e^{ { \cos ^{ -1 }{ \sin { \left( x+\pi /3 \right) } } } }$, then
Question 14 :
The function $\cfrac { \left| x \right| }{ { x }^{ 2 }+2x } , x\ne 0$ and $f(0)=0$ is not continuous at $x=0$ because-
Question 15 :
lf $f(x)=\displaystyle \begin{cases}\dfrac{a^{2[x]+\{x\}}-1}{2[x]+\{x\}};x\neq 0 \\ \log a;x=0 \end{cases}$ where $[.\ ]$ and $\{.\ \}$ denote integral and fractional part respectively, then<br/>
Question 16 :
If the function $f(x)$ defined as$\displaystyle f\left( x \right) =\begin{cases} \begin{matrix} { \left( \sin { x } +\cos { x }  \right)  }^{ \csc { x }  }, & -\frac { \pi  }{ 2 } <x<0 \end{matrix} \\ \begin{matrix} a, & x=0 \end{matrix} \\ \begin{matrix} \frac { { e }^{ 1/x }+{ e }^{ 2/x }+{ e }^{ 3/x } }{ a{ e }^{ -2+1/x }+b{ e }^{ -1+3/x } } , & 0<x<\frac { \pi  }{ 2 }  \end{matrix} \end{cases}$ is continuous at $x=0$, then
Question 17 :
The value of $f\left ( 0 \right )$ so that the function$\displaystyle f\left ( x \right )=\frac{1-\cos \left ( 1-\cos x \right )}{x^{4}}$is continuous everywhere is
Question 18 :
If $f(x)=\begin{cases} \cfrac { { x }^{ 2 }-(a+2)x+2a }{ x-2 } ,\quad \quad x\neq 2 \\ 2,\quad \quad \quad \quad \quad \quad \quad x=2 \end{cases}$ is continuous at $x=2$, then $a$ is equal to-
Question 19 :
Let $f(x) = \left\{\begin{matrix} -2,& -3 \leq x \leq 0\\ x - 2,  & x < x \leq 3\end{matrix}\right.$ and $g(x) = f(|x|) + |f(x)|$<br/>Which of the following statements are correct?<br/>1. $g(x)$ is continuous at $x = 0$.<br/>2. $g(x)$ is continuous at $x = 2$.<br/>3. $g(x)$ is continuous at $x = -1$.<br/>Select the correct answer using the code given below
Question 20 :
If$\displaystyle f\left ( x \right )=\frac{\sin 3x+A\sin 2x+B\sin x}{x^{5}},x\neq 0$, iscontinous at $x = 0$ then<br>
Question 21 :
lf the function $\displaystyle \mathrm{f}({x})=\frac{e^{x^{2}}-\cos {x}}{x^{2}}$ for $x \neq 0$ is continuous at ${x}=0$ then $\mathrm{f}(\mathrm{0})=$<br/>
Question 23 :
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 24 :
The function$f\left ( x \right )=1+\left | \cos x \right |$
Question 25 :
Let $[x]$ denote the integral part of $\displaystyle x\in R,\> g(x)=x-[x]$. Let $f(x)$ be any continuous function with $f(0)=f(1)$, then the function $h(x)=f(g(x))$:
Question 26 :
The function $y = f(x)$ is defined by $x = 2t - |t|, y =t^2+t |t|, t\in R$ in the interval $x\in [-1,1]$ then<br>
Question 27 :
<br>lf $f(x)=<br>\left\{\begin{matrix}(1+|\sin x|)^{\displaystyle \frac{a}{|\sin x|}}&-\displaystyle \frac{\pi}{6}<x<0\\b&x=0 \\e^{\displaystyle \frac{\tan 2x}{\tan 3x}} &0<x<\displaystyle \frac{\pi}{6}\end{matrix}\right.$ is<br><br>continuous at $\mathrm{x}=0$ then<br>
Question 28 :
If f is defined by $f(x) = \left\{\begin{matrix} x, for \ 0 \le x < 1  \\ 2 - x, for \ x \ge 1\end{matrix}\right.$ , then at $X = 1$, is Discuss the nature of the function
Question 29 :
The functions $f(x) = \left (\dfrac {\log_{e}(1 + ax) - \log_{e}(1 - bx)}{x}\right )$ is undefined at $x = 0$. The value which should be assigned to $f$ at $x = 0$ so that it is continuous at $x = 0$ is
Question 30 :
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>
