Question

question
evaluate the limit: $lim_{x
ightarrow12}\frac{x - 12}{sqrt{x + 13}-5}$
answer

Explanation:

Step1: Rationalize the denominator

Multiply numerator and denominator by $\sqrt{x + 13}+5$.
\[

$$\begin{align*} &\lim_{x ightarrow12}\frac{x - 12}{\sqrt{x + 13}-5}\times\frac{\sqrt{x + 13}+5}{\sqrt{x + 13}+5}\\ =&\lim_{x ightarrow12}\frac{(x - 12)(\sqrt{x + 13}+5)}{(x + 13)-25}\\ =&\lim_{x ightarrow12}\frac{(x - 12)(\sqrt{x + 13}+5)}{x - 12} \end{align*}$$

\]

Step2: Simplify the expression

Cancel out the common factor $(x - 12)$.
\[

$$\begin{align*} &\lim_{x ightarrow12}\frac{(x - 12)(\sqrt{x + 13}+5)}{x - 12}\\ =&\lim_{x ightarrow12}(\sqrt{x + 13}+5) \end{align*}$$

\]

Step3: Evaluate the limit

Substitute $x = 12$ into $\sqrt{x + 13}+5$.
\[

$$\begin{align*} &\sqrt{12+13}+5\\ =&\sqrt{25}+5\\ =&5 + 5\\ =&10 \end{align*}$$

\]

Answer:

10