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One-sided limits and existence of a two-sided limit

Consider the function f(x)=∣x∣xf(x) = \dfrac{|x|}{x} defined for all x≠0x \neq 0.

Examine the behavior of f(x)=∣x∣xf(x) = \dfrac{|x|}{x} as xx approaches 00 from the left and from the right.

One-sided limits and existence of a two-sided limit graph

Using the graph (or by testing values of xx close to 00 on each side), determine lim⁡x→0−f(x)\displaystyle\lim_{x\to 0^-} f(x) and lim⁡x→0+f(x)\displaystyle\lim_{x\to 0^+} f(x), and state whether lim⁡x→0f(x)\displaystyle\lim_{x\to 0} f(x) exists. If it exists, give its value; if not, explain why not.

Hint

Simplify ∣x∣/x|x|/x separately for x<0x<0 and for x>0x>0 using the definition of absolute value, then compare the two resulting constant values.

Step-by-step solution

For x<0x < 0, we have ∣x∣=−x|x| = -x, so f(x)=−xx=−1f(x) = \dfrac{-x}{x} = -1 for every negative xx. Thus as x→0−x \to 0^-, the function stays exactly at −1-1, so lim⁡x→0−f(x)=−1\displaystyle\lim_{x\to 0^-} f(x) = -1.

For x>0x > 0, we have ∣x∣=x|x| = x, so f(x)=xx=1f(x) = \dfrac{x}{x} = 1 for every positive xx. Thus lim⁡x→0+f(x)=1\displaystyle\lim_{x\to 0^+} f(x) = 1. Since the left-hand limit (−1-1) and right-hand limit (11) are not equal, the two-sided limit lim⁡x→0f(x)\displaystyle\lim_{x\to 0} f(x) does not exist.

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