$\mathop {\lim }\limits_{x \to a} \frac{{{x^2} - {a^2}}}{{x - a}} = $

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 $f: R \rightarrow R$ is defined by $f(x) = [x-3] + |x-4|$ for $x \in R$,then $\lim_{x \rightarrow 3^{-}} f(x)$ is equal to

$\lim _{n \rightarrow \infty} n\left(\sqrt{n^2+9}-n\right)=$

$\lim _{x \rightarrow 0} \frac{\sqrt{1+x \sin x}-\sqrt{\cos x}}{\tan ^2 2 x}=$

Let $f(x) = \lim_{y \to 0} \frac{(1 - \cos(xy))\tan(xy)}{y^3}$. Then the number of solutions of the equation $f(x) = \sin x, x \in R$ is: