We are asked to evaluate the limit of the function $\frac{(x-1)^2}{1-x^2}$ as $x$ approaches 1.

1. Problem Description

We are asked to evaluate the limit of the function $\frac{(x-1)^2}{1-x^2}$ as $x$ approaches

1.

2. Solution Steps

First, we can factor the denominator:

$1 - x^2 = (1 - x)(1 + x) = -(x - 1)(x + 1)$

Therefore, the expression becomes:

$\lim_{x \to 1} \frac{(x - 1)^2}{1 - x^2} = \lim_{x \to 1} \frac{(x - 1)^2}{-(x - 1)(x + 1)}$

Now we can cancel out a factor of $(x - 1)$ from the numerator and denominator, since $x \neq 1$ as $x$ approaches 1:

$\lim_{x \to 1} \frac{(x - 1)}{-(x + 1)}$

Now, we can substitute $x = 1$ into the expression:

$\frac{(1 - 1)}{-(1 + 1)} = \frac{0}{-2} = 0$

3. Final Answer

The final answer is 0.