$\lim _{x \rightarrow 0} \frac{\sin x \sin ^{-1} x}{x^2}$ is equal to

$\lim _{x}$ ${\rightarrow \infty} \frac{(2 x+1)^{50}+(2 x+2)^{50}+(2 x+3)^{50}+\cdots \cdots+(2 x+100)^{50}}{(2 x)^{50}+(10)^{50}} = \dots$

Let $t_{n}$ denote the $n^{th}$ term of the infinite series $\frac{1}{1 !} + \frac{10}{2 !} + \frac{21}{3 !} + \frac{34}{4 !} + \frac{49}{5 !} + \ldots$. Then $\lim _{n \rightarrow \infty} t_{n}$ is

$\mathop {\lim }\limits_{n \to \infty } {({3^n} + {4^n})^{\frac{1}{n}}} = $

If $f(x) = \begin{cases} |x|+1, & x < 0 \\ 0, & x = 0 \\ |x|-1, & x > 0 \end{cases}$,for what value$(s)$ of $a$ does $\lim_{x \to a} f(x)$ exist?

$\lim _{x \rightarrow \infty} x\left(\log \left(1+\frac{x}{2}\right)-\log \frac{x}{2}\right) = $