question\nevaluate the limit: $lim_{x\rightarrow12}\frac{x - 12}{sqrt{x + 13}-5}$\nanswer
question\nevaluate the limit: $lim_{x\rightarrow12}\frac{x - 12}{sqrt{x + 13}-5}$\nanswer
Answer
Answer:
10
Explanation:
Step1: Rationalize the denominator
Multiply numerator and denominator by $\sqrt{x + 13}+5$. [ \begin{align*} &\lim_{x\rightarrow12}\frac{x - 12}{\sqrt{x + 13}-5}\times\frac{\sqrt{x + 13}+5}{\sqrt{x + 13}+5}\ =&\lim_{x\rightarrow12}\frac{(x - 12)(\sqrt{x + 13}+5)}{(x + 13)-25}\ =&\lim_{x\rightarrow12}\frac{(x - 12)(\sqrt{x + 13}+5)}{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 + 13}+5)}{x - 12}\ =&\lim_{x\rightarrow12}(\sqrt{x + 13}+5) \end{align*} ]
Step3: Evaluate the limit
Substitute $x = 12$ into $\sqrt{x + 13}+5$. [ \begin{align*} &\sqrt{12+13}+5\ =&\sqrt{25}+5\ =&5 + 5\ =&10 \end{align*} ]