$\lim _{x \rightarrow 0} \frac{(1-e^x) \sin x}{x^2+x^3}$ is equal to

Let $l = \mathop {\lim}\limits_{x \to 0} \frac{[x]^2}{x^2}$ and $m = \mathop {\lim}\limits_{x \to 0} \frac{[x^2]}{x^2}$,where $[ \cdot ]$ denotes the greatest integer function. Then:

If $\mathop {\lim }\limits_{x \to \infty } \frac{e^{\mu x} + 5}{e^{100x} + 7}$ exists,then the sum of all possible positive integral values of $\mu$ is:

$\mathop {\lim }\limits_{x \to \infty } \,{\left( {\frac{{x + a}}{{x + b}}} \right)^{x + b}} = $

For each $x \in \mathbb{R}$,let $[x]$ be the greatest integer less than or equal to $x$. Then $\lim_{x \to 0^+} \frac{x([x] + |x|) \sin [x]}{|x|}$ is equal to

If $a > 0$ and $n \in R$,then $\lim_{x \rightarrow a} x^n = \dots$