Problem 39 - Precalculus: Mathematics for Calculus

Question

Find the limit, if it exists. If the limit does not exist, explain why. $$\lim _{x \rightarrow 2} \frac{|x-2|}{x-2}$$

Step-by-Step Solution

Answer

The limit does not exist because the one-sided limits are not equal.

Step 1: Analyze One-Sided Limits (Approaching from the Right)

Let's analyze the limit as \(x\) approaches 2 from the right, i.e., \(x \rightarrow 2^+\). When \(x\) is slightly greater than 2, \(|x-2| = x-2\). Therefore, the expression becomes \(\frac{x-2}{x-2} = 1\). Thus, \(\lim_{x \rightarrow 2^+} \frac{|x-2|}{x-2} = 1\).

Step 2: Analyze One-Sided Limits (Approaching from the Left)

Now, let's analyze the limit as \(x\) approaches 2 from the left, i.e., \(x \rightarrow 2^-\). When \(x\) is slightly less than 2, \(|x-2| = -(x-2)\). Therefore, the expression becomes \(\frac{-(x-2)}{x-2} = -1\). Thus, \(\lim_{x \rightarrow 2^-} \frac{|x-2|}{x-2} = -1\).

Step 3: Compare the One-Sided Limits

For the limit \(\lim_{x \rightarrow 2} \frac{|x-2|}{x-2}\) to exist, the one-sided limits must be equal. From Step 1, we found \(\lim_{x \rightarrow 2^+} \frac{|x-2|}{x-2} = 1\) and from Step 2, we found \(\lim_{x \rightarrow 2^-} \frac{|x-2|}{x-2} = -1\). Since these two values are not equal, the limit does not exist.

Key Concepts

One-Sided Limits

When dealing with limits, especially in calculus, understanding **one-sided limits** is crucial. A one-sided limit examines what happens to a function as the input gets infinitely close to a point from one side—either the left or the right. In essence, it checks the behavior of a function as it approaches a specific point.

• The right-hand limit (denoted \[\lim_{x \rightarrow a^+} f(x)\] ) involves approaching the point \(a\) from values greater than \(a\).
• The left-hand limit (denoted \(\lim_{x \rightarrow a^-} f(x)\)) involves approaching the point \(a\) from values less than \(a\).

For limits to exist generally, both the left-hand and right-hand limits must not only exist but also be equal. If they differ, the overall limit does not exist.

Absolute Value

The **absolute value** function, represented by |x|, is a distance measurement that denotes how far a number is from zero on the number line, without considering direction. This provides non-negative values regardless of whether the original input is positive or negative.

• For positive values of \(x\), \(|x| = x\).
• For negative values of \(x\), \(|x| = -x\).

When evaluating limits involving absolute values, such as \(\lim _{x \rightarrow a} \frac{|x-a|}{x-a}\), the absolute value function alters the expression based on whether \(x\) is greater or less than \(a\). This is why the expression behaves differently as \(x\) approaches from left and right sides.

Limit Existence

In calculus, the question of **limit existence** asks whether a function approaches a specific value as the input approaches some point. For a limit to exist at a particular point:

• Both one-sided limits (from the left and the right) must exist.
• The one-sided limits must be equal—if they aren't, the overall limit does not exist.

Exploring the example \(\lim _{x \rightarrow 2} \frac{|x-2|}{x-2}\), we see that as \(x\) approaches 2 from the left, the limit is -1; but from the right, it is 1. This difference indicates that no single number represents the behavior of the function around \(x = 2\), hence the limit does not exist at this point.