Problem 29 - Precalculus: Mathematical for Calculus

Question

Find the limit, if it exists. If the limit does not exist, èxplain 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: Understand the function

Consider the expression \( \frac{|x-2|}{x-2} \). This is a piece-wise function because the absolute value introduces different expressions depending on whether \( x-2 \) is positive or negative.

Step 2: Consider the limit from the right

As \( x \to 2^+ \), \( x > 2 \) and \( |x-2| = x-2 \). Therefore, \( \frac{|x-2|}{x-2} = \frac{x-2}{x-2} = 1 \). Thus, \( \lim _{x \to 2^+} \frac{|x-2|}{x-2} = 1 \).

Step 3: Consider the limit from the left

As \( x \to 2^- \), \( x < 2 \) and \( |x-2| = -(x-2) \). Therefore, \( \frac{|x-2|}{x-2} = \frac{-(x-2)}{x-2} = -1 \). Thus, \( \lim _{x \to 2^-} \frac{|x-2|}{x-2} = -1 \).

Step 4: Compare one-sided limits

The limit from the right is 1 and from the left is -1. Since these one-sided limits are not equal, the overall limit \( \lim _{x \rightarrow 2} \frac{|x-2|}{x-2} \) does not exist.

Key Concepts

Piecewise Functions

A piecewise function is a mathematical expression defined by multiple sub-functions, each applying to a certain interval or "piece" of the function's domain. It allows different expressions to be used at various input values. In the exercise provided, the function \( \frac{|x-2|}{x-2} \) is divided into two expressions based on the value of \( x \) relative to 2. This happens because the absolute value expression behaves differently above and below 2:

• If \( x > 2 \), then \( |x-2| = x-2 \), simplifying the function to 1.
• If \( x < 2 \), then \( |x-2| = -(x-2) \), simplifying the function to -1.

Thus, this piecewise nature helps handle functions that change behavior at specific points, enabling us to evaluate complex limits through separate analysis of each segment.

One-Sided Limits

One-sided limits focus on understanding the behavior of a function as it approaches a specific point from one side—either from the right or the left. When calculating the overall limit, these values are crucial. For functions like the one in the problem, evaluating one-sided limits helps capture the variation in outputs:

• Approaching from the right (\( x \to 2^+ \)), when \( x > 2 \), the limit simplifies to \( \,1 \).
• Approaching from the left (\( x \to 2^- \)), when \( x < 2 \), the limit simplifies to \(-1 \).

These calculations show that the right-hand and left-hand limits differ, indicating the overall limit at \( x = 2 \) does not exist. This difference is critical for understanding piecewise and discontinuous functions, as it highlights jumps or breaks in their graphs.

Absolute Value Functions

Absolute value functions are expressed as \( |x| \) and are unique because they consider only the magnitude, not the direction, of a number on the number line. This makes absolute expressions show no negativity in their outcomes. The function provided in our exercise is \( |x-2| \), and such expressions adjust based on the value of \( x \):

• When \( x \geq 2 \), \( |x-2| = x-2 \), revealing the expression's linear behavior.
• When \( x < 2 \), \( |x-2| = -(x-2) \), adjusting the slope and ensuring the absolute value is positive.

This switching mechanism is pivotal in calculus when dealing with discontinuities and the behavior of functions around certain critical points. Understanding this function's behavior helps in constructing and analyzing more complex expressions where the absolute value impacts limits and continuity.