Limits and Cont...

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

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

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-

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

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 

$$\displaystyle\lim_{x\rightarrow \dfrac{\pi}{2}}\dfrac{\cot x-\cos x}{(\pi -2x)^3}$$ equals?

For $$x>y$$, $$\displaystyle\lim_{x\rightarrow 0}{\left[\left(\sin{x}\right)^{1/x}+\left(\cfrac{1}{x}\right)^{\sin{x}}\right]}$$ is :

If the function $$f(x)$$ satisfies the relation $$f(x+y)=y\dfrac{|x-1|}{(x-1)}f(x)+f(y)$$ with $$f(1)=2$$, then $$\displaystyle\lim_{x\rightarrow 1}f'(x)$$ is?