← Back to learning objective

One-sided limits of the sign function and existence of the two-sided limit

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

The function f(x)=x∣x∣f(x) = \dfrac{x}{|x|} is graphed below.

One-sided limits of the sign function and existence of the two-sided limit graph

Using the graph (or by evaluating f(x)f(x) for values of xx close to 00 from each side), determine lim⁡x→0−f(x)\lim_{x \to 0^-} f(x) and lim⁡x→0+f(x)\lim_{x \to 0^+} f(x). Then state whether lim⁡x→0f(x)\lim_{x \to 0} f(x) exists, and if it does not exist, explain why in one sentence.

Hint

Recall that ∣x∣=−x|x| = -x when x<0x < 0 and ∣x∣=x|x| = x when x>0x > 0. Simplify f(x)f(x) separately for xx slightly less than 00 and xx slightly greater than 00, then compare the two one-sided limits.

Step-by-step solution

For x<0x < 0, we have ∣x∣=−x|x| = -x, so f(x)=x−x=−1f(x) = \dfrac{x}{-x} = -1 for every negative xx. Since f(x)=−1f(x) = -1 for all xx near 00 from the left, the left-hand limit is lim⁡x→0−f(x)=−1\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, giving lim⁡x→0+f(x)=1\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)\lim_{x \to 0} f(x) does not exist.

Reasoning about why the two-sided limit fails to exist

Hint
Step-by-step solution

The right-hand limit equals one, while the left-hand limit equals negative one. Since one is greater than negative one, the right-hand limit is larger. This mismatch between the two one-sided limits is exactly why the two-sided limit does not exist.

Related problems

Practice similar Calculus problems interactively.

Start Closing the Gap