Question

question
evaluate the limit: $$\lim_{x \to 12} \frac{x - 12}{\sqrt{x - 8} - 2}$$

Explanation:

Step1: Rationalize the denominator

Multiply numerator and denominator by $\sqrt{x - 8}+2$.

$$\lim_{x ightarrow12}\frac{(x - 12)(\sqrt{x - 8}+2)}{(\sqrt{x - 8}-2)(\sqrt{x - 8}+2)}$$

Step2: Simplify the denominator

Use the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$. Here $a=\sqrt{x - 8}$ and $b = 2$.

$$\lim_{x ightarrow12}\frac{(x - 12)(\sqrt{x - 8}+2)}{(x - 8)-4}=\lim_{x ightarrow12}\frac{(x - 12)(\sqrt{x - 8}+2)}{x - 12}$$

Step3: Cancel out the common factor

Cancel out the common factor $(x - 12)$ (since $x
eq12$ as we are taking the limit as $x
ightarrow12$).

$$\lim_{x ightarrow12}(\sqrt{x - 8}+2)$$

Step4: Substitute $x = 12$

Substitute $x = 12$ into the expression $\sqrt{x - 8}+2$.
$$\sqrt{12 - 8}+2=\sqrt{4}+2=2 + 2$$

Answer:

$4$