Q.Find $\lim_{x\to0^-}\dfrac{|x|}{x}$ and $\lim_{x\to0^+}\dfrac{|x|}{x}$. Does $\lim_{x\to0}\dfrac{|x|}{x}$ exist?

For $x<0$, $|x|=-x$, so $\dfrac{|x|}{x}=\dfrac{-x}{x}=-1$ for every negative $x$, giving $\lim_{x\to0^-}\dfrac{|x|}{x}=-1$. For $x>0$, $|x|=x$, so $\dfrac{|x|}{x}=\dfrac{x}{x}=1$ for every positive $x$, giving $\lim_{x\to0^+}\dfrac{|x|}{x}=1$. Since $-1\neq1$, the left-hand and right-hand limits disagree, so by the definition of Section 1, $\lim_{x\ …

Use the piecewise definition of $|x|$ separately for $x<0$ and $x>0$ to evaluate each one-sided limit, then compare …

Assuming the two-sided limit exists and equals $1$ (checking only the right-hand side); confusing $|x|/x$ …