Let $l = \mathop {Lim}\limits_{x \to {0^ + }} x^m (\ln x)^n$ where $m, n \in N$,then:

$\lim _{x \rightarrow 1} \frac{\log x}{1-x} = $

If $f(x)$ is a differentiable function,then $\mathop {\lim }\limits_{x \to a} \frac{af(x) - xf(a)}{x - a}$ is

Let $f : R \rightarrow R$ be a differentiable function such that $f \left(\frac{\pi}{4}\right)=\sqrt{2}$,$f \left(\frac{\pi}{2}\right)=0$ and $f^{\prime}\left(\frac{\pi}{2}\right)=1$. Let $g(x)=\int\limits_{x}^{\pi / 4}\left(f^{\prime}(t) \sec t+\tan t \sec t f(t)\right) d t$ for $x \in\left[\frac{\pi}{4}, \frac{\pi}{2}\right)$. Then $\lim\limits _{ x \rightarrow\left(\frac{\pi}{2}\right)^{-}} g ( x )$ is equal to

The $\mathop {\lim }\limits_{x \to 0} {(\cos x)^{\cot x}}$ is

$\mathop {\lim }\limits_{x \to a} \frac{{\cos x - \cos a}}{{\cot x - \cot a}} = $