Problem 58 - A Graphical Approach to Precalculus with Limits

Question

Evaluate each limit. (a) \(\lim _{x \rightarrow 4^{-}} \frac{x-4}{|x-4|}\) (b) \(\lim _{x \rightarrow 4^{+}} \frac{x-4}{|x-4|}\) (c) \(\lim _{x \rightarrow 4} \frac{x-4}{|x-4|}\)

Step-by-Step Solution

Answer

(a) -1, (b) 1, (c) does not exist.

Step 1: Understanding Part (a)

For part (a), we consider the limit as \(x\) approaches 4 from the left, denoted as \(x \rightarrow 4^-\). For values of \(x\) that are slightly less than 4, \(x-4\) is negative, so \(|x-4| = -(x-4)\). The expression becomes \(\frac{x-4}{|x-4|} = \frac{x-4}{-(x-4)} = -1\).

Step 2: Compute the Limit for Part (a)

Since the expression simplifies to -1 as \(x\) approaches 4 from the left, the limit is simply \(-1\). Thus, \(\lim_{x \rightarrow 4^-} \frac{x-4}{|x-4|} = -1\).

Step 3: Understanding Part (b)

For part (b), we consider the limit as \(x\) approaches 4 from the right, denoted as \(x \rightarrow 4^+\). For values of \(x\) that are slightly greater than 4, \(x-4\) is positive, so \(|x-4| = x-4\). The expression becomes \(\frac{x-4}{|x-4|} = \frac{x-4}{x-4} = 1\).

Step 4: Compute the Limit for Part (b)

Since the expression simplifies to 1 as \(x\) approaches 4 from the right, the limit is simply \(1\). Thus, \(\lim_{x \rightarrow 4^+} \frac{x-4}{|x-4|} = 1\).

Step 5: Understanding Part (c)

For part (c), we must consider both the left and right-hand limits as \(x\) approaches 4. From the previous results, the left-hand limit is -1 and the right-hand limit is 1. For the two-sided limit to exist, both one-sided limits must be equal.

Step 6: Compute the Limit for Part (c)

Since the left-hand limit and the right-hand limit are not equal (\(-1 eq 1\)), the limit does not exist. Therefore, \(\lim_{x \rightarrow 4} \frac{x-4}{|x-4|}\) does not exist.

Key Concepts

One-Sided Limits

In calculus, a one-sided limit focuses on what happens to a function as its input approaches a particular value from only one direction — either the left or the right. This is particularly useful when examining functions that may behave differently depending on the direction from which the variable approaches a certain point.

When dealing with one-sided limits, you might see notation like \( x \to c^- \) or \( x \to c^+ \). This notation is used to indicate whether you are approaching the limit from the left (minus) or the right (plus). If you consider the limit expressed as \( \lim_{x \to 4^-} \), it represents the value the function approaches as \(x\) approaches 4 from values less than 4. Similarly, \( \lim_{x \to 4^+} \) represents the approach from values greater than 4.

One-sided limits are crucial when analyzing piecewise functions or functions involving division by zero or absolute values, as they provide insight into potential discontinuities in the function.

Absolute Value

The concept of absolute value in mathematics is about the distance of a number from zero on the number line. It is always non-negative. For any real number \( x \), the absolute value is denoted by \( |x| \).

In function terms, the absolute value function splits into two cases:

• If \( x \geq 0 \), then \( |x| = x \).
• If \( x < 0 \), then \( |x| = -x \).

This means the absolute value function behaves differently depending on whether its argument is positive or negative, making it essential to examine the sign of the expression inside the absolute value when determining limits.

For instance, when evaluating \( \frac{x-4}{|x-4|} \), you need to establish whether \( x-4 \) is positive or negative. This will affect the simplification needed to find the limit and helps explain behavioral changes in the function at the critical point \( x = 4 \).

Limit Does Not Exist

Limits are a fundamental concept in calculus used to describe the behavior of functions as they approach a particular point. A limit may not exist if a function does not approach a single, definitive value as \( x \) approaches a given point.

There are several scenarios where a limit might not exist:

• The function approaches different values from the left and right (disjoint left and right-hand limits).
• The function approaches infinity or negative infinity.
• The function exhibits rapid oscillation near the point of interest.

For the given problem, the limit as \( x \) approaches 4 does not exist because the one-sided limits from the left and right are not equal. Specifically, \( \lim_{x \to 4^-} = -1 \) and \( \lim_{x \to 4^+} = 1 \). These differing results indicate that there isn't a single value that \( \frac{x-4}{|x-4|} \) approaches as \( x \) nears 4. Thus, we say the limit does not exist at this point.