This is the limit expression:

$\lim_{x \to 1} \frac{1 - \sqrt{x}}{1 - x^2}$

Let me solve this step by step:

Step 1: Factorize the denominator

The denominator $1 - x^2$ can be factored as: $1 - x^2 = (1 - x)(1 + x)$

So the expression becomes: $\frac{1 - \sqrt{x}}{(1 - x)(1 + x)}$

Step 2: Handle indeterminate form

Substituting $x = 1$ directly gives an indeterminate form $\frac{0}{0}$. Therefore, we need to simplify further.

Step 3: Rationalize the numerator

Multiply the numerator and denominator by the conjugate of $1 - \sqrt{x}$, which is $1 + \sqrt{x}$. This gives: $\frac{(1 - \sqrt{x})(1 + \sqrt{x})}{(1 - x)(1 + x)(1 + \sqrt{x})}$

Using the difference of squares in the numerator: $1 - \sqrt{x^2} = 1 - x$

So the expression becomes: $\frac{1 - x}{(1 - x)(1 + x)(1 + \sqrt{x})}$

Step 4: Cancel the common factor $1 - x$

For $x \neq 1$, cancel $1 - x$ from the numerator and denominator: $\frac{1}{(1 + x)(1 + \sqrt{x})}$

Step 5: Substitute $x = 1$

Now substitute $x = 1$ directly: $\frac{1}{(1 + 1)(1 + \sqrt{1})} = \frac{1}{2 \cdot 2} = \frac{1}{4}$

Final Answer:

$\lim_{x \to 1} \frac{1 - \sqrt{x}}{1 - x^2} = \frac{1}{4}$

Let me know if you have any questions or need a detailed explanation!

Here are five related questions to explore further:

1. What is the importance of rationalizing the numerator in limits?
2. How does factoring help simplify indeterminate forms?
3. Can L'Hôpital's Rule be used to solve this problem? How?
4. How do you determine when to apply algebraic simplifications in limits?
5. What are some other common techniques for solving limits involving square roots?

Tip: Always check for alternative approaches like factoring or rationalizing when dealing with indeterminate forms in limits.