Question

$\underset{x\to 0}{\lim }\left[\frac{\sin x}{x}\right]$, where [.] denotes the greatest integer function

• A. is equal to 1
• B. is equal to zero
• C. does not exists
• D. none of these

Answer 1

Answer: (B)

See the graph of $y=x$ and $\sin x$ in the following figure

From the graph, when $x\to 0^{+}$, the graph of $y=x$ is above the graph of $y=\sin x$, i.e.,
$\sin \dot{x}<x\Rightarrow \frac{\sin x}{x}<1$
$\Rightarrow \underset{x\to 0^{+}}{\lim }\left[\frac{\sin x}{x}\right]=0$
When $x\to 0^{-}$, 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 $\underset{x\to 0^{-}}{\lim }\left[\frac{\sin x}{x}\right]=0$
Thus, $\underset{x\to 0}{\lim }\left[\frac{\sin x}{x}\right]=0$