Question

The value of \(\lim_{x \to 1^-} \frac{x-1}{|x^2-1|}\) is given by

• A. \(\frac{1}{2}\)
• B. -\(\frac{1}{2}\)
• C. -1
• D. 1

Hint:

Always establish the sign of the terms inside absolute value bars based on the direction of the limit approach (\( x \to a^- \) or \( x \to a^+ \)) before simplifying the algebraic fraction.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This problem requires evaluating a left-sided limit containing an absolute value expression in the denominator.
Evaluating limits with absolute values requires removing the absolute value bars by determining the sign of the inside expression within the interval of interest.

Step 2: Detailed Explanation:
Let us analyze the limit expression:
\[ L = \lim_{x \to 1^-} \frac{x-1}{|x^2-1|} \] - The notation \( x \to 1^- \) means that \( x \) approaches \( 1 \) from the left, meaning \( x \) is slightly less than \( 1 \) (\( x < 1 \)).
- Let us determine the sign of the expression inside the absolute value bars, \( x^2 - 1 \), when \( x < 1 \) (and \( x \) is close to 1):
Since \( x < 1 \), squaring both sides (since \( x \) is positive near 1) gives:
\[ x^2 < 1 \implies x^2 - 1 < 0 \] - Since the expression is negative, we remove the absolute value bars by multiplying the expression by \( -1 \):
\[ |x^2 - 1| = -(x^2 - 1) = 1 - x^2 \] - Now, substitute this simplified expression back into the limit:
\[ L = \lim_{x \to 1^-} \frac{x-1}{1-x^2} \] - Factor the denominator using the difference of squares identity \( 1 - x^2 = (1-x)(1+x) \):
\[ L = \lim_{x \to 1^-} \frac{x-1}{(1-x)(1+x)} \] Notice that \( x-1 = -(1-x) \). Substitute this into the numerator:
\[ L = \lim_{x \to 1^-} \frac{-(1-x)}{(1-x)(1+x)} \] Cancel out the common term \( (1-x) \) from both the numerator and the denominator (which is valid since \( x \neq 1 \)):
\[ L = \lim_{x \to 1^-} \frac{-1}{1+x} \] - Now, substitute \( x = 1 \) directly into this simplified expression:
\[ L = \frac{-1}{1+1} = -\frac{1}{2} \]

Step 3: Final Answer:
The value of the limit is \( -\frac{1}{2} \).
Therefore, the correct choice is Option (B).