Limits and Deri...

Find the derivative of $\sqrt{\tan x}$ with respect to $x$ using the first principle.

$\underset { x\rightarrow 0 }{ lim } \frac { tan(sinx)-x }{ { tanx }^{ 3 } }$ is equal to 

If $\overset { \pi  }{ \underset { 0 }{\int  }  } x \, f(sin\,x)dx=A\overset { \pi /2 }{ \underset { 0 }{ \int }  } f(sin\,x)dx,$ then A is _____________.

The value of $\displaystyle n\xrightarrow { lim } \infty\frac{1.n+2.(n-1)+3.(n-2)+...+n.1}{{1}^{2}+{2}^{2}+...+{n}^{2}}$ is

The value of $\displaystyle \lim _{ x\rightarrow 0 } \dfrac{1+\sin{x}-\cos{x}+\log{(1-x)}}{x^{3}}$, is

The value of $\displaystyle \lim_{\theta \rightarrow 0^{+}} \dfrac {\sin \sqrt {\theta}}{\sqrt {\sin  \theta}}$ is equal to

$\displaystyle\lim _{ x\rightarrow 0 }{ { x }^{ 2 }{ e }^{ \sin { \frac { 1 }{ x }  }  } }$ equals 

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

Let $f:(0, \infty)\to R$ be a differentiable function such that $f'(x)=2-\dfrac{f(x)}{x}$ for all $x\in (0, \infty)$ and $f(1)\neq 1$. Then 

If $\mathrm { L } = \lim _ { \mathrm { x } ^ { 2 } \rightarrow \mathrm { a } } \frac { \mathrm { b } - \cos \left( \mathrm { x } ^ { 2 } - \mathrm { a } \right) } { \left( \mathrm { x } ^ { 2 } - \mathrm { a } \right) \sin \left( \mathrm { cx } ^ { 2 } - \mathrm { a } \right) }$ is non-
zero finite $( \mathrm { a } > 0 ) ,$ then-