$\mathop {\lim }\limits_{x \to 0} \cos \frac{1}{x}$

Evaluate the limit: $\mathop {\lim }\limits_{x \to \infty } [x({a^{1/x}} - 1)]$,where $a > 1$.

$\lim _{x \rightarrow 2}\left(\frac{5 x-8}{8-3 x}\right)^{\frac{3}{2 x-4}} = $

$\mathop {\lim }\limits_{x \to 0} \sin \left( {\frac{1}{x}} \right)$ is

If $f(x) = \begin{cases} \frac{\sin(1+[x])}{[x]}, & \text{for } [x] \neq 0 \\ 0, & \text{for } [x] = 0 \end{cases}$ where $[x]$ denotes the greatest integer function,then $\lim_{x \rightarrow 0^{-}} f(x)$ is equal to

$\lim _{x \rightarrow \infty} \frac{3 x+4 \cos ^2 x}{\sqrt{x^2-5 \sin ^2 x}} = $