Question

question evaluate the limit: $lim_{x
ightarrow - 3}\frac{sqrt{x + 7}-2}{-5x - 15}$ answer

Explanation:

Step1: Rationalize the numerator

Multiply the fraction by $\frac{\sqrt{x + 7}+2}{\sqrt{x + 7}+2}$.
\[

$$\begin{align*} &\lim_{x ightarrow - 3}\frac{\sqrt{x + 7}-2}{-5x - 15}\times\frac{\sqrt{x + 7}+2}{\sqrt{x + 7}+2}\\ =&\lim_{x ightarrow - 3}\frac{(x + 7)-4}{(-5x - 15)(\sqrt{x + 7}+2)}\\ =&\lim_{x ightarrow - 3}\frac{x + 3}{-5(x + 3)(\sqrt{x + 7}+2)} \end{align*}$$

\]

Step2: Simplify the fraction

Cancel out the common factor $(x + 3)$ (since $x
eq - 3$ when taking the limit).
\[

$$\begin{align*} &\lim_{x ightarrow - 3}\frac{x + 3}{-5(x + 3)(\sqrt{x + 7}+2)}\\ =&\lim_{x ightarrow - 3}\frac{1}{-5(\sqrt{x + 7}+2)} \end{align*}$$

\]

Step3: Substitute $x=-3$

\[

$$\begin{align*} &\frac{1}{-5(\sqrt{-3 + 7}+2)}\\ =&\frac{1}{-5(2 + 2)}\\ =&-\frac{1}{20} \end{align*}$$

\]

Answer:

$-\frac{1}{20}$