The given function is:

$f(x) = \frac{x^2 - 1}{\sqrt{x} - 1}$

We are asked to find the limit:

$\lim_{x \to 1} f(x)$

Step-by-Step Solution

First, substitute $x = 1$ directly into the function to check for any undefined form:

$f(1) = \frac{1^2 - 1}{\sqrt{1} - 1} = \frac{1 - 1}{1 - 1} = \frac{0}{0}$

This results in an indeterminate form $\frac{0}{0}$, so we must apply algebraic techniques to simplify the expression before finding the limit.

Step 1: Factor the numerator

The numerator $x^2 - 1$ is a difference of squares, which can be factored as:

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

Now, the function becomes:

$f(x) = \frac{(x - 1)(x + 1)}{\sqrt{x} - 1}$

Step 2: Rationalize the denominator

To simplify the expression, multiply both the numerator and the denominator by the conjugate of the denominator, which is $\sqrt{x} + 1$:

$f(x) = \frac{(x - 1)(x + 1)}{\sqrt{x} - 1} \cdot \frac{\sqrt{x} + 1}{\sqrt{x} + 1}$

Simplifying the denominator:

$(\sqrt{x} - 1)(\sqrt{x} + 1) = x - 1$

Now the expression becomes:

$f(x) = \frac{(x - 1)(x + 1)(\sqrt{x} + 1)}{x - 1}$

Step 3: Cancel the common factor

Since $x - 1$ is present in both the numerator and the denominator, we can cancel it out (as long as $x \neq 1$):

$f(x) = (x + 1)(\sqrt{x} + 1)$

Step 4: Substitute $x = 1$

Now substitute $x = 1$ into the simplified expression:

$f(1) = (1 + 1)(\sqrt{1} + 1) = 2 \times 2 = 4$

Thus, the limit is:

$\lim_{x \to 1} f(x) = 4$

Final Answer:

$\lim_{x \to 1} f(x) = 4$

Do you need further details or have any questions about this solution? Here are five related questions to consider:

1. How does the difference of squares help in solving limits?
2. What is the importance of rationalizing the denominator in limit problems?
3. Can L'Hopital's Rule be used here to solve the indeterminate form? Why or why not?
4. How would you handle this limit if it involved a cube root instead of a square root?
5. What is the difference between a limit existing and a function being defined at a certain point?

Tip: Always check if the limit leads to an indeterminate form before applying algebraic manipulations or calculus techniques.