question\nevaluate the limit: $$\\lim_{x \\to 12} \\frac{x - 12}{\\sqrt{x - 8} - 2}$$
question\nevaluate the limit: $$\\lim_{x \\to 12} \\frac{x - 12}{\\sqrt{x - 8} - 2}$$
Answer
Explanation:
Step1: Rationalize the denominator
Multiply numerator and denominator by $\sqrt{x - 8}+2$. $$\lim_{x\rightarrow12}\frac{(x - 12)(\sqrt{x - 8}+2)}{(\sqrt{x - 8}-2)(\sqrt{x - 8}+2)}$$
Step2: Simplify the denominator
Use the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$. Here $a=\sqrt{x - 8}$ and $b = 2$. $$\lim_{x\rightarrow12}\frac{(x - 12)(\sqrt{x - 8}+2)}{(x - 8)-4}=\lim_{x\rightarrow12}\frac{(x - 12)(\sqrt{x - 8}+2)}{x - 12}$$
Step3: Cancel out the common factor
Cancel out the common factor $(x - 12)$ (since $x\neq12$ as we are taking the limit as $x\rightarrow12$). $$\lim_{x\rightarrow12}(\sqrt{x - 8}+2)$$
Step4: Substitute $x = 12$
Substitute $x = 12$ into the expression $\sqrt{x - 8}+2$. $$\sqrt{12 - 8}+2=\sqrt{4}+2=2 + 2$$
Answer:
$4$