prove that $\lim_{x\to 0^{-}}\frac{x}{|x|}$

Question

Question

How do I prove that $\displaystyle\lim_{x\to 0^{-}}\frac{x}{|x|}=-1$?

Because I thought that the answer would be DNE?

Answer 1

Notice that

$$\frac{x}{|x|}=\begin{cases}-1,\;\;x<0\\ \\+1,\;\;x>0\end{cases}$$

Also, $x\to 0^-$ is notation for $x$ converging to $0$ from the left.