The given mathematical expression is:

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

We can simplify this limit using algebraic manipulation. Specifically, we can use the technique of multiplying by the conjugate. Here are the steps:

Step-by-Step Solution:

1. Multiply by the conjugate: Multiply both the numerator and the denominator by the conjugate of the numerator, $\sqrt{x} + 2$:

$\frac{\sqrt{x} - 2}{4 - x} \cdot \frac{\sqrt{x} + 2}{\sqrt{x} + 2}$

2. Simplify the numerator: $(\sqrt{x} - 2)(\sqrt{x} + 2) = x - 4$

Now the expression becomes:

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

3. Cancel out the common factor $4 - x$ in the numerator and the denominator:

$\frac{-1}{\sqrt{x} + 2}$

4. Evaluate the limit as $x \to 4$:

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

Final Answer:

$\boxed{-\frac{1}{4}}$

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:

1. How do you apply the conjugate method to limits involving square roots?
2. What other techniques can you use for solving limits when direct substitution results in an indeterminate form?
3. How does this limit relate to L'Hopital's Rule, and could it be applied here?
4. Can you explain how limits involving radicals differ from polynomial limits?
5. What are the conditions for canceling terms in limit problems?

Tip: Always check if a function can be simplified algebraically before considering more advanced techniques like L'Hopital's Rule.