Let $[x]$ denote the greatest integer less than or equal to $x$ for any real number $x$. Then,$\lim _{n \rightarrow \infty} \frac{[n \sqrt{2}]}{n}$ is equal to

The limit of $x \sin \left(e^{\frac{1}{x}}\right)$ as $x \rightarrow 0$

$\lim _{x \rightarrow \infty}\left(\frac{2+\sin x}{x^2+3}\right)$ is equal to

Let $f: R \rightarrow (0, \infty)$ be a strictly increasing function such that $\lim _{x \rightarrow \infty} \frac{f(7 x)}{f(x)}=1$. Then,the value of $\lim _{x \rightarrow \infty} \left[\frac{f(5 x)}{f(x)}-1\right]$ is equal to

$\mathop {\lim }\limits_{x \to \infty } \frac{{\sin x}}{x} = $

Let $f: (0, \infty) \rightarrow \mathbb{R}$ and $g: (0, \infty) \rightarrow \mathbb{R}$ be two functions where $g(x) = x + \frac{1}{x}$. If $1 < f(x) \cdot g(x) < 10$ for all $x > 0$,then $\lim_{x \to \infty} f(x)$ is