Existence of Limits


In Exercises 5 and 6, explain why the limits do not exist.


limx→0 x/|x|

Verified step by step guidance

1. To determine the existence of the limit \( \lim_{x \to 0} \frac{x}{|x|} \), we need to consider the behavior of the function \( \frac{x}{|x|} \) as \( x \) approaches 0 from both the left and the right.
2. First, consider the limit as \( x \to 0^+ \) (approaching 0 from the right). For \( x > 0 \), the absolute value function \( |x| \) is simply \( x \). Therefore, \( \frac{x}{|x|} = \frac{x}{x} = 1 \).
3. Next, consider the limit as \( x \to 0^- \) (approaching 0 from the left). For \( x < 0 \), the absolute value function \( |x| \) is \( -x \). Therefore, \( \frac{x}{|x|} = \frac{x}{-x} = -1 \).
4. Since the limit from the right (\( x \to 0^+ \)) is 1 and the limit from the left (\( x \to 0^- \)) is -1, the two one-sided limits are not equal.
5. Because the one-sided limits are not equal, the overall limit \( \lim_{x \to 0} \frac{x}{|x|} \) does not exist. For a limit to exist at a point, the left-hand limit and the right-hand limit must be equal.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points of discontinuity. The limit can exist or not exist depending on the values the function approaches from different directions.

One-Sided Limits

One-sided limits refer to the limits of a function as the input approaches a specific value from one side only, either the left (denoted as lim x→c-) or the right (denoted as lim x→c+). For the limit to exist at a point, both one-sided limits must be equal. If they differ, the overall limit does not exist.

Absolute Value Function

The absolute value function, denoted as |x|, outputs the non-negative value of x regardless of its sign. This function creates a piecewise definition, which can lead to different behaviors when approaching zero from the left or right. Understanding how the absolute value affects limits is crucial for analyzing functions that involve it.