Limit and discontinuity of x/|x|

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Compute $\displaystyle\lim_{x\to 0}\frac{x}{|x|}$ and classify the type of discontinuity of $f(x)=\dfrac{x}{|x|}$ at $x=0$.

Solution

Let us make the absolute value explicit: $\frac{x}{|x|} = \begin{cases} \dfrac{x}{x}=1 & x>0 \\[4pt] \dfrac{x}{-x}=-1 & x<0 \end{cases}$ The one-sided limits are therefore $\lim_{x\to 0^+}\frac{x}{|x|}=1, \qquad \lim_{x\to 0^-}\frac{x}{|x|}=-1.$ Since they are different, the limit $\displaystyle\lim_{x\to 0}\frac{x}{|x|}$ $\boxed{\text{does not exist}}$.

Since the two one-sided limits are finite but different, at $x=0$ there is a discontinuity of the first kind (jump), of amplitude $|1-(-1)|=2$. (It is the “sign” function, $\operatorname{sgn}(x)$.)

Links

Topics: Continuity
Concepts: First-kind discontinuity · Right and left limits · Absolute value
Skills: Calculating limits · Reasoning by cases
Exercise types: Computing a limit