question evaluate the limit: $lim_{x\rightarrow - 3}\frac{sqrt{x + 7}-2}{-5x - 15}$ answer
question evaluate the limit: $lim_{x\rightarrow - 3}\frac{sqrt{x + 7}-2}{-5x - 15}$ answer
Answer
Explanation:
Step1: Rationalize the numerator
Multiply the fraction by $\frac{\sqrt{x + 7}+2}{\sqrt{x + 7}+2}$. [ \begin{align*} &\lim_{x\rightarrow - 3}\frac{\sqrt{x + 7}-2}{-5x - 15}\times\frac{\sqrt{x + 7}+2}{\sqrt{x + 7}+2}\ =&\lim_{x\rightarrow - 3}\frac{(x + 7)-4}{(-5x - 15)(\sqrt{x + 7}+2)}\ =&\lim_{x\rightarrow - 3}\frac{x + 3}{-5(x + 3)(\sqrt{x + 7}+2)} \end{align*} ]
Step2: Simplify the fraction
Cancel out the common factor $(x + 3)$ (since $x\neq - 3$ when taking the limit). [ \begin{align*} &\lim_{x\rightarrow - 3}\frac{x + 3}{-5(x + 3)(\sqrt{x + 7}+2)}\ =&\lim_{x\rightarrow - 3}\frac{1}{-5(\sqrt{x + 7}+2)} \end{align*} ]
Step3: Substitute $x=-3$
[ \begin{align*} &\frac{1}{-5(\sqrt{-3 + 7}+2)}\ =&\frac{1}{-5(2 + 2)}\ =&-\frac{1}{20} \end{align*} ]
Answer:
$-\frac{1}{20}$