Q55 / 100 · BITSAT 2023 (Memory Based Paper 1) · Mathematics
$\lim _{x \rightarrow 0} \frac{|\sin x|}{x}$ is equal to
- A1
- B-1
- Correct answerCDoes not exist
- DNone of these
Solution
Let $l=\lim _{x \rightarrow 0} \frac{[\sin x]}{x} ; \mathrm{RHL}=\lim _{x \rightarrow 0^{+}} \frac{\sin x}{x}=1$
$\mathrm{LHL}=\lim _{x \rightarrow 0^{-}}\left(\frac{-\sin x}{x}\right)=-\left(\lim _{x \rightarrow 0^{-}} \frac{\sin x}{x}\right)=-1$
$\left[|x|=\begin{array}{c}x \text { if } x \geq 0 \\ -x \text { if } x \leq 0\end{array}\right]$
As, LHL $\neq$ RHL Hence, limit $l$.does not exist.
$\mathrm{LHL}=\lim _{x \rightarrow 0^{-}}\left(\frac{-\sin x}{x}\right)=-\left(\lim _{x \rightarrow 0^{-}} \frac{\sin x}{x}\right)=-1$
$\left[|x|=\begin{array}{c}x \text { if } x \geq 0 \\ -x \text { if } x \leq 0\end{array}\right]$
As, LHL $\neq$ RHL Hence, limit $l$.does not exist.
Hence, the correct answer is option: 'C'.