Computing Limits — Question 5

Question 5 - Solution

We are given: $$\lim_{x \to 4} \frac{x - 4}{\sqrt{x} - 2}$$

Step 1: Direct Substitution

Substitute $x = 4$: $$\frac{4 - 4}{\sqrt{4} - 2} = \frac{0}{0}$$

This is an indeterminate form, so we need to simplify.

Step 2: Use Algebraic Manipulation

We can rationalize the denominator. Multiply the numerator and denominator by the conjugate of the denominator:

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

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

So the expression becomes: $$\frac{(x - 4)(\sqrt{x} + 2)}{x - 4}$$

Cancel the common factor $x - 4$ (note: valid as $x \to 4$, not at $x = 4$): $$= \sqrt{x} + 2$$

Now take the limit: $$\lim_{x \to 4} \sqrt{x} + 2 = \sqrt{4} + 2 = 2 + 2 = \boxed{4}$$