Find the limit: $\mathop {\lim }\limits_{x \to 2} \left[\frac{x^{3}-4 x^{2}+4 x}{x^{2}-4}\right]$

If $f(x) = \text{Sgn}(\text{Sgn}(\text{Sgn}(x)))$,then $\mathop {\lim }\limits_{x \to 0} f(x)$ is equal to :-

Consider the following statements:
Statement $1$: $\lim _{x \rightarrow 1} \frac{a x^{2}+b x+c}{c x^{2}+b x+a} = 1$ (where $a+b+c \neq 0$).
Statement $2$: $\lim _{x \rightarrow -2} \frac{\frac{1}{x}+\frac{1}{2}}{x+2} = \frac{1}{4}$.

The value of $\mathop {\lim }\limits_{x \to 0} \frac{1}{x}\left[ {{{\tan }^{ - 1}}\left( {\frac{{x + 1}}{{2x + 1}}} \right) - \frac{\pi }{4}} \right]$ is

Let $m$ and $n$ be two positive integers greater than $1$. If $\lim_{\alpha \rightarrow 0} \left( \frac{e^{\cos(\alpha^n)} - e}{\alpha^m} \right) = -\left( \frac{e}{2} \right)$,then the value of $\frac{m}{n}$ is

Evaluate the given limit: $\mathop {\lim }\limits_{x \to 0} x \sec x$