Problem 37 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find \(\lim _{x \rightarrow 1}\left(x^{2}-1\right) /|x-1|\) or state that it does not exist.

Step-by-Step Solution

Answer

The limit does not exist because the left and right limits are not equal.

Step 1: Analyze the Expression

We need to find \( \lim_{x \to 1} \frac{x^2 - 1}{|x-1|} \). First, let's simplify the expression \(x^2 - 1\) which factors to \((x-1)(x+1)\). The expression becomes \(\frac{(x-1)(x+1)}{|x-1|}\).

Step 2: Consider Left-Hand Limit

For \(x \to 1^-\) (approaching from the left), \(x-1\) is negative, so \(|x-1| = -(x-1)\). The expression becomes \(- (x+1)\). Evaluate \(\lim_{x \to 1^-} -(x+1) = -2\).

Step 3: Consider Right-Hand Limit

For \(x \to 1^+\) (approaching from the right), \(x-1\) is positive, so \(|x-1| = x-1\). The expression simplifies to \(x+1\). Evaluate \(\lim_{x \to 1^+} (x+1) = 2\).

Step 4: Compare the Two Limits

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

Key Concepts

Right-Hand Limit

When working with limits, understanding the right-hand limit is crucial, especially near points of discontinuity or critical points. The right-hand limit of a function as it approaches a particular value means that we are considering values of the function as the input approaches the given point from the right.

In mathematical terms, for a function \( f(x) \), the right-hand limit as \( x \) approaches \( c \) from the right is represented as \( \lim_{x \to c^+} f(x) \). This notation implies that we are only interested in the values of \( f(x) \) where \( x > c \).

For instance, considering our exercise, we have examined the right-hand limit of \( \frac{(x-1)(x+1)}{|x-1|} \) as \( x \to 1^+ \). Since \( x-1 \) is positive, \( |x-1| = x-1 \). Therefore, the expression reduces to \( x+1 \), and its limit equals 2.

This aspect highlights how the sign and behavior of the expression change when approaching from the right.

Left-Hand Limit

The left-hand limit is another fundamental concept in understanding the behavior of functions as they approach specific points. It analyzes the values of a function as the input comes closer from the left.

In notation, if \( f(x) \) is our function, the left-hand limit as \( x \) approaches \( c \) is written as \( \lim_{x \to c^-} f(x) \). This means that we look at values where \( x < c \).

In the provided example, we analyze the left-hand limit of \( \frac{(x-1)(x+1)}{|x-1|} \) as \( x \to 1^- \). Here, \( x-1 \) is negative, making \( |x-1| = -(x-1) \). The expression becomes \( -(x+1) \), and the left-hand limit evaluates to -2.

Understanding the left-hand limit is essential because it shows how functions behave differently from either side of a critical value.

Piecewise Functions

Piecewise functions are fascinating constructs in mathematics, often defined by different expressions based on subsets of the domain. This means a piecewise function can have multiple rules, each applicable to certain intervals of the domain.

They are very useful in real-world applications where different conditions apply over different ranges, like tax brackets or physics problems involving different states of matter.

In the context of the exercise, although not explicitly a piecewise function, the limit can be treated similarly because the behavior of the function changes depending on whether \( x \) approaches 1 from the left or the right. Specifically, the expression \( \frac{(x-1)(x+1)}{|x-1|} \) can be treated as having two separate behaviors based on the sign of \( x-1 \):

• For \( x > 1 \) (right-hand side), use \( x+1 \).
• For \( x < 1 \) (left-hand side), use \( -(x+1) \).

This gives insight into how piecewise-like analysis can be applied, offering a deeper understanding of limits and function behavior.