$\lim _{x \rightarrow 0} \left( \frac{\tan x}{\sqrt{2x+4}-2} \right)$ is equal to

If $f(x) = \cos x$ when $x = n\pi$ $(n = 0, 1, 2, 3, \dots)$ and $f(x) = 3$ otherwise,and $\phi(x) = \begin{cases} x^2 + 1 & \text{when } x \neq 3, x \neq 0 \\ 3 & \text{when } x = 0 \\ 5 & \text{when } x = 3 \end{cases}$,then find $\lim_{x \to 0} f(\phi(x))$.

$\mathop {\lim }\limits_{x \to \infty } {\left[ {1 + \frac{1}{{mx}}} \right]^x}$ is equal to

Given below are two statements:
Statement $I$: $\lim _{x \rightarrow 0} \left( \frac{\tan ^{-1} x + \log _e \sqrt{\frac{1+x}{1-x}} - 2x}{x^5} \right) = \frac{2}{5}$
Statement $II$: $\lim _{x \rightarrow 1} \left( x^{\frac{2}{1-x}} \right) = \frac{1}{e^2}$
In the light of the above statements,choose the correct answer from the options given below:

$\lim _{x \rightarrow 0}(1+3x)^{\frac{2}{x}} = $

$\mathop {\lim }\limits_{n \to \infty } {({4^n} + {5^n})^{1/n}}$ is equal to