Question

Find the limit: $\underset{x\to 0}{\lim }\frac{\sin \left(\left|x\right|\right)}{x}$.

Answer 1

Compute the required limit:

For $x<0,\left|x\right|=-x$

$$\begin{gathered} \Rightarrow \underset{x\to 0^{-}}{\lim }\frac{\sin \left(\left|x\right|\right)}{x} \\ \Rightarrow \underset{x\to 0^{-}}{\lim }\frac{\sin \left(-x\right)}{x} \\ \Rightarrow \underset{x\to 0^{-}}{\lim }\frac{-\sin \left(x\right)}{x} \\ \Rightarrow -1 \end{gathered}$$ {Since, $\sin (-x)=-\sin \left(x\right)$ and $\underset{x\to 0}{\lim }\left(\frac{\sin x}{x}\right)=1$}

For $x>0,\left|x\right|=x$

$$\begin{gathered} \Rightarrow \underset{}{\underset{x\to 0^{+}}{\lim }\frac{\sin \left(\left|x\right|\right)}{x}} \\ \Rightarrow \underset{x\to 0^{-}}{\lim }\frac{\sin \left(x\right)}{x} \\ \Rightarrow 1 \end{gathered}$$

Since, the two limits from the left and from the right are different.

Therefore the above limit does not exist.