$\lim _{x \rightarrow \frac{\pi}{4}} \frac{4 \sqrt{2}-(\cos x+\sin x)^5}{1-\sin 2 x} = $

Evaluate $\mathop {\lim }\limits_{x \to {1^ + }} \frac{{{{\left( {1 + \left\{ x \right\}} \right)}^{\frac{1}{{\left\{ x \right\}}}}} - \frac{e}{{\sqrt {{e^{\left\{ x \right\}}}} }}}}{{1 - \cos \left\{ x \right\}}}$ (where $\{.\}$ denotes the fractional part function).

$\lim _{x \rightarrow 1} (\log _3 3x)^{\log _x 8} = \ldots$

Let $f(x)$ be differentiable at $x = h$. Then $\lim_{x \to h} \frac{(x + h)f(x) - 2hf(h)}{x - h}$ is equal to

Given that $f'(2) = 6$ and $f'(1) = 4$,then $\mathop {\lim }\limits_{h \to 0} \frac{{f(2h + 2 + {h^2}) - f(2)}}{{f(h - {h^2} + 1) - f(1)}} = $

Let $f(x) = \int_0^x (t + \sin(1 - e^t)) dt, x \in R$. Then $\lim_{x \rightarrow 0} \frac{f(x)}{x^3}$ is equal to