evaluate the limit: $lim_{x\rightarrow12}\frac{x - 12}{sqrt{x - 8}-2}$
evaluate the limit: $lim_{x\rightarrow12}\frac{x - 12}{sqrt{x - 8}-2}$
Answer
Answer:
4
Explanation:
Step1: Rationalize the denominator
Multiply numerator and denominator by $\sqrt{x - 8}+2$. [ \begin{align*} &\lim_{x\rightarrow12}\frac{x - 12}{\sqrt{x - 8}-2}\times\frac{\sqrt{x - 8}+2}{\sqrt{x - 8}+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} \end{align*} ]
Step2: Simplify the expression
Cancel out the common factor $(x - 12)$. [ \begin{align*} &\lim_{x\rightarrow12}\frac{(x - 12)(\sqrt{x - 8}+2)}{x - 12}\ =&\lim_{x\rightarrow12}(\sqrt{x - 8}+2) \end{align*} ]
Step3: Evaluate the limit
Substitute $x = 12$ into $\sqrt{x - 8}+2$. [ \begin{align*} &\sqrt{12 - 8}+2\ =&\sqrt{4}+2\ =&2 + 2\ =&4 \end{align*} ]