Introduction

The limit of an absolute value function often involves determining how the function behaves as the input approaches a particular point, especially around points where the expression inside the absolute value changes sign.

Key Concept

Often limits involving absolute value do not exist. For example: $$\lim_{x\to0}\dfrac{x}{|x|}.$$ Because of the discontinuity on the graph of $y=\dfrac{x}{|x|}$, this limit does not exist.
When dealing with limits of absolute value functions, you may need to consider one-sided limits to handle the behavior on both sides of the point of interest.

Example

Let’s find the limit:

$$\lim_{x \to 3} |x – 3|$$

Similar steps apply here:
1. Left-hand limit:
$$\lim_{x \to 3^-} |x – 3| = \lim_{x \to 3^-} -(x – 3) = 0$$

2. Right-hand limit:
$$\lim_{x \to 3^+} |x – 3| = \lim_{x \to 3^+} (x – 3) = 0$$

Again, since both limits are equal, we have:
$$\lim_{x \to 3} |x – 3| = 0$$

Example with a Non-zero Result

For a different approach, consider

$$\lim_{x \to 3} |x^2 – 1|$$

In the proximity of $x=3$ the expression inside the absolute value is positive, thus we can drop the absolute value lines:

$$\lim_{x \to 3} |x^2 – 1| = 3^2 – 1 = 8$$

Alternatively, if we were asked to find the limit:
$$\lim_{x \to 0} |x^2 – 1|$$

Now, in the proximity of $x=0$ the expression inside the absolute value is negative, thus we will remove the absolute value and switch the sign of each term of the expression inside.

$$\lim_{x \to 0} |x^2 – 1| = -0^2 + 1 = 1$$

Conclusion

When finding limits of the expressions with absolute value, consider breaking them into cases based on where the expression inside becomes positive or negative.