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Q.
$\displaystyle\lim _{x \rightarrow 0}\left[\frac{\sin x}{x}\right]$, where [.] denotes the greatest integer function
Limits and Derivatives
Solution:
See the graph of $y=x$ and $\sin x$ in the following figure
From the graph, when $x \rightarrow 0^{+}$, the graph of $y=x$ is above the graph of $y=\sin x$, i.e.,
$\sin \dot{x}< x \Rightarrow \frac{\sin x}{x}< 1 $
$\Rightarrow \displaystyle\lim _{x \rightarrow 0^{+}}\left[\frac{\sin x}{x}\right]=0$
When $x \rightarrow 0^{-}$, the graph of $y=x$ is below the graph of $y=\sin x$, i.e.,
$\sin x >x \Rightarrow \frac{\sin x}{x}< 1$ (As $x$ is negative)
or $\displaystyle \lim _{x \rightarrow 0^{-}}\left[\frac{\sin x}{x}\right]=0$
Thus, $\displaystyle\lim _{x \rightarrow 0}\left[\frac{\sin x}{x}\right]=0$
