The quadratic equation whose roots are $m$ and $n$,where $m = \lim_{x \rightarrow 0} \frac{x \log(1+2x)}{x \tan x}$ and $n = \lim_{x \rightarrow 0} \frac{\log x + \log(\frac{1+x}{x})}{x}$,is

$ABC$ is an isosceles triangle inscribed in a circle of radius $r$. If $AB = AC$ and $h$ is the altitude from $A$ to $BC$,and $P$ is the perimeter of $ABC$,then $\mathop {\lim }\limits_{h \to 0} \frac{\Delta }{{{P^3}}}$ equals (where $\Delta$ is the area of the triangle).

If $\mathop {\lim }\limits_{n \to \infty } n \cos \left( \frac{\pi }{4n} \right) \sin \left( \frac{\pi }{4n} \right) = k$,then $k$ is equal to

$\lim _{x \rightarrow 0} \frac{x^4+x^3+x^2}{\sin ^{-1}\left(\frac{x}{\sqrt{1+x^2}}\right) \cdot \tan ^{-1} x} = $

Evaluate: $\mathop {\lim }\limits_{x \to 1} \frac{x^{15}-1}{x^{10}-1}$

$\lim _{n \rightarrow \infty} \frac{1}{n^3+1}+\frac{4}{n^3+1}+\frac{9}{n^3+1}+\ldots+\frac{n^2}{n^3+1} = $