Limits and Cont...

If function $$f(x)=\begin{cases} \begin{matrix} \dfrac{\sin 3x}{x}; & x \ne 0 \end{matrix} \\ \begin{matrix} m; & x=0 \end{matrix} \\ \begin{matrix}  &  \end{matrix} \end{cases}$$
is continuous at $$x=2$$ then value of $$m$$ will be:

The value of $$\displaystyle \lim_{x\rightarrow 0} \dfrac {xe^{x} - \log_{e} (1 + x)}{x^{2}}$$ is

The value of $$\displaystyle \lim_{x\rightarrow \infty} \dfrac {\sin x}{x}$$ is

The value of $$\displaystyle \lim_{x\rightarrow 0} \dfrac {1 - \cos x}{x^{2}}$$ is

The value of $$\displaystyle \lim_{x\rightarrow 0} \left (\dfrac {\sin 3x}{\tan x}\right )^{4}$$ is

The value of $$\displaystyle \lim_{x\rightarrow \infty} \sin \dfrac {\pi}{4x} \cos \dfrac {\pi}{4x}$$ is

Let $$\alpha(a)$$ and $$\beta(a)$$ be the roots of the equation $$(\sqrt[3]{1+a}-1)x^{2}+(\sqrt{1+a}-1){x}+(\sqrt[6]{1+a}-1)=0$$ where $$a>-1$$. Then $$ \underset{a\rightarrow 0^{+}}{\lim}\alpha(a)$$ and $$ \underset{a\rightarrow 0^{+}}{\lim}\beta(a)$$ are 

Let $$f(x)=\displaystyle \frac{(256+ax)^{1/8}-2}{(32+bx)^{1/5}-2}$$. If $$f$$ is continuous at $$x = 0$$, then the value of $$a / b$$ is:

If $$f(x)=\left\{\begin{matrix}(cos x)^{1/sinx} &for &x\neq 0 \\  k & for & x=0 \end{matrix}\right.$$
Then the value of $$k$$, so that $$f$$ is continuous at $$x=0$$ is

The function $$f$$ : $$R/\{0\}\rightarrow R$$ given by $$f(x)=\displaystyle \frac{1}{x}-\frac{2}{e^{2x}-1}$$ can be made continuous at $$x=0$$ by defining $$f(0)$$ as -