$\mathop {\lim }\limits_{x \to \pi /2} \frac{\tan 3x}{x} = $

Evaluate $\mathop {\lim }\limits_{x \to 0} f(x),$ where $f(x) = \begin{cases} \frac{|x|}{x}, & x \neq 0 \\ 0, & x=0 \end{cases}$

If $\lim_{x \to 0} \frac{(4^x - 1)^3}{\tan(\frac{x}{4}) \log(1 + \frac{x^2}{3})} = 96(\log a)^b$, then $(a + b) = $

If $f(x) = \begin{cases} x, & \text{when } 0 \le x \le 1 \\ 2 - x, & \text{when } 1 < x \le 2 \end{cases}$,then $\lim_{x \to 1} f(x) = $

If $f(x) = \frac{1-x+\sqrt{9x^2+10x+1}}{2x}$,then $\lim_{x \rightarrow -1^{-}} f(x) = $

The quadratic polynomial $p(x)$ has roots $1$ and $\alpha$, while quadratic polynomial $q(x)$ has roots $1$ and $\beta$. Let $\alpha$ and $\beta$ be the roots of $r(x) = p(x) + q(x)$. Then $\lim_{x \to \infty} [\sqrt{p(x)} - \sqrt{q(x)}] = $