Question

Let \[ f(x) = \frac{x^2 - 1}{|x| - 1}. \] \(\text{Then the value of}\) \[ \lim_{x \to 1} f(x) \text{ is:} \]

• A. -1
• B. 1
• C. 2
• D. 3

Hint:

When dealing with absolute value functions, simplify the expression by considering the behavior of \( |x| \) in different intervals, and then apply standard limit techniques.

Answer 1

The Correct Option is C

We are given the function: \[ f(x) = \frac{x^2 - 1}{|x| - 1}. \] We are asked to find the limit of \( f(x) \) as \( x \) approaches 1. Step 1: Simplify the expression We first notice that \( x^2 - 1 = (x - 1)(x + 1) \), so we can rewrite the function as: \[ f(x) = \frac{(x - 1)(x + 1)}{|x| - 1}. \] Now, we examine the behavior of the function near \( x = 1 \). Step 2: Evaluate the limit For \( x \to 1 \), we need to consider the behavior of \( |x| - 1 \). Since \( |x| = x \) for \( x > 0 \), we have: \[ |x| - 1 = x - 1 \text{for} x > 1. \] Substituting this into the function: \[ f(x) = \frac{(x - 1)(x + 1)}{x - 1}. \] For \( x \neq 1 \), we can cancel out the \( (x - 1) \) term: \[ f(x) = x + 1. \] Now, take the limit as \( x \to 1 \): \[ \lim_{x \to 1} f(x) = 1 + 1 = 2. \] Thus, the value of the limit is \( \boxed{2} \).