To evaluate the limit $\lim_{x \to 1} \frac{\sqrt{5x - 4} - \sqrt{x}}{x - 1}$, we'll use algebraic manipulation again. Here are the steps:

1. Identify the indeterminate form: As $x \to 1$, both the numerator and the denominator approach 0, creating a $\frac{0}{0}$ indeterminate form.

2. Rationalize the numerator: Multiply and divide by the conjugate of the numerator $\sqrt{5x - 4} - \sqrt{x}$, which is $\sqrt{5x - 4} + \sqrt{x}$.

$\lim_{x \to 1} \frac{\sqrt{5x - 4} - \sqrt{x}}{x - 1} \cdot \frac{\sqrt{5x - 4} + \sqrt{x}}{\sqrt{5x - 4} + \sqrt{x}}$

This transforms the expression into:

$\lim_{x \to 1} \frac{(\sqrt{5x - 4} - \sqrt{x})(\sqrt{5x - 4} + \sqrt{x})}{(x - 1)(\sqrt{5x - 4} + \sqrt{x})}$

3. Simplify the numerator: Using the difference of squares formula, $a^2 - b^2 = (a - b)(a + b)$, we get:

$\lim_{x \to 1} \frac{(5x - 4) - x}{(x - 1)(\sqrt{5x - 4} + \sqrt{x})}$

This simplifies to:

$\lim_{x \to 1} \frac{4x - 4}{(x - 1)(\sqrt{5x - 4} + \sqrt{x})}$

4. Factor the numerator: The numerator $4x - 4$ can be factored as $4(x - 1)$:

$\lim_{x \to 1} \frac{4(x - 1)}{(x - 1)(\sqrt{5x - 4} + \sqrt{x})}$

5. Cancel common factors: The $(x - 1)$ terms cancel out:

$\lim_{x \to 1} \frac{4}{\sqrt{5x - 4} + \sqrt{x}}$

6. Evaluate the limit: Substitute $x = 1$:

$\frac{4}{\sqrt{5(1) - 4} + \sqrt{1}} = \frac{4}{\sqrt{1} + 1} = \frac{4}{1 + 1} = \frac{4}{2} = 2$

So, the limit is:

$\boxed{2}$

Would you like more details or have any questions?

Here are 5 more questions you might find interesting:

1. How do you solve limits using L'Hôpital's Rule?
2. What is the derivative of $\sqrt{5x - 4}$?
3. How do you find the limit of a function involving trigonometric expressions?
4. Can you explain the Squeeze Theorem with an example?
5. What are the applications of limits in real-life problems?

Tip: When faced with indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$, consider using algebraic manipulation or L'Hôpital's Rule to simplify the expression before evaluating the limit.