Question

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

Short answer

The limit does not exist.

Step-by-step solution

Set up the expression for limits from both sides

We need to evaluate the limit from both directions (from the left and from the right) because of the absolute value expression in the denominator. Therefore, write the limit as \( \lim_{x \to 1^-} \frac{x^2 - 1}{|x-1|} \) and \( \lim_{x \to 1^+} \frac{x^2 - 1}{|x-1|} \).

Simplify the numerator

Observe that \( x^2 - 1 \) can be factored as \( (x-1)(x+1) \). This simplification will be used in both left-hand limit and right-hand limit expressions.

Evaluate the left-hand limit

For \( x < 1 \), \( |x - 1| = -(x - 1) \). Thus the limit becomes \( \lim_{x \to 1^-} \frac{(x-1)(x+1)}{-(x-1)} \). The \( (x-1) \) terms cancel out, leaving us with \( \lim_{x \to 1^-} -(x+1) = -2 \).

Evaluate the right-hand limit

For \( x > 1 \), \( |x - 1| = x - 1 \). Thus the limit becomes \( \lim_{x \to 1^+} \frac{(x-1)(x+1)}{x-1} \). The \( (x-1) \) terms cancel out, leaving us with \( \lim_{x \to 1^+} (x+1) = 2 \).

Compare the left-hand and right-hand limits

Since \( \lim_{x \to 1^-} \frac{x^2 - 1}{|x-1|} = -2 \) and \( \lim_{x \to 1^+} \frac{x^2 - 1}{|x-1|} = 2 \), the two one-sided limits are not equal. Therefore, the limit \( \lim_{x \to 1} \frac{x^2 - 1}{|x-1|} \) does not exist.

Key concepts

Left-hand limit

The concept of a left-hand limit allows us to understand the behavior of a function as it approaches a specific point from the left side. For example, when evaluating the left-hand limit of a function at a point, we analyze what happens to the outputs as the inputs approach the point slightly less than the target value. This is denoted by the notation \( \lim_{x \to a^-} f(x) \), where the minus sign indicates the approaching value from the left.

In our exercise, as we compute \( \lim_{x \to 1^-} \frac{x^2 - 1}{|x-1|} \), we utilize the fact that for \( x < 1 \), the expression \( |x - 1| \) simplifies to \(-(x - 1)\). This adjustment is essential because the absolute value flips the sign of the distance when \( x \) is less than the pivot point. After simplifying and cancelling terms, we find the left-hand limit to have a distinct value of \(-2\).

This approach helps determine the behavior of a function in one particular direction, relevant in understanding real-life contexts where continuity or sudden changes are critical.

Right-hand limit

The right-hand limit is quite similar to the left-hand limit, but it evaluates the function's behavior as it approaches a specific point from the right side. Symbolically, it's represented as \( \lim_{x \to a^+} f(x) \), with the plus sign showing the right-hand approach.

In our exercise, determining \( \lim_{x \to 1^+} \frac{x^2 - 1}{|x-1|} \) involves recognizing that for \( x > 1 \), the absolute value \( |x - 1| \) retains its original form, i.e., \( x - 1 \). Simplifying by cancelling terms, the right-hand limit resolves to \(2\).

Studying the right-hand limit is just as crucial since analyzing the behavior from either direction provides insights into any potential discontinuities or asymmetrical behaviors around the pinpointed value. This analysis reveals dynamic changes as one transitions from left to right past a threshold, such as a physical barrier or threshold level in real-world applications.

Absolute value

The absolute value function is a key mathematical concept that transforms any given number into its non-negative counterpart. In notation, it uses vertical bars, such as \(|x|\), which conveys the distance of \(x\) from zero on the number line.

The expression within the absolute value will yield the same result whether the input is positive or negative by negating any negative entries. In mathematical contexts like limits, understanding how absolute values impact the function's behavior is crucial, especially near the points where changes in direction occur.

In our exercise, the term \(|x-1|\) notably becomes critical as we approach the point \(x=1\). For values of \(x\) less than 1, \(|x-1|\) flips the sign (\(-(x-1)\)), while for values greater than 1, it simply retains the term \(x-1\). Knowing how to manage this transformation allows for a clearer interpretation of changes within expressions and helps in solving problems involving discontinuities or sharp transitions in values.

One-sided limits

One-sided limits enhance our understanding of how functions behave as they approach a particular value, analyzing either from just below or just above the target. They are critical when examining functions with sudden jumps or discontinuities.

This concept is prominently applied in our exercise as we consider both \( \lim_{x \to 1^-} \) and \( \lim_{x \to 1^+} \). Note that even though the overall limit doesn’t exist at \(x = 1\) due to differing one-sided limits, these individual one-sided limit values still provide valuable insights.

In mathematics, understanding one-sided limits can aid in functions that aren’t perfectly smooth or continuous, making it vital for scenarios such as material fatigue in engineering or threshold behaviors in economics. One-sided limits reveal the intricacies of functions that might not be evident when solely considering the two-sided limit, giving a much-needed detailed perspective.