evaluate the limit, if it exists. (if an answer does not exist, enter dne.)
lim_{x
ightarrow - 24}\frac{sqrt{x^{2}+49}-25}{x + 24}

Question

evaluate the limit, if it exists. (if an answer does not exist, enter dne.)
 lim_{x
ightarrow - 24}\frac{sqrt{x^{2}+49}-25}{x + 24}

Accepted Answer

# Explanation:
## Step1: Rationalize the numerator
Multiply by $\frac{\sqrt{x^{2}+49}+25}{\sqrt{x^{2}+49}+25}$.
$$
\begin{align*}
&\lim_{x
ightarrow - 24}\frac{\sqrt{x^{2}+49}-25}{x + 24}\times\frac{\sqrt{x^{2}+49}+25}{\sqrt{x^{2}+49}+25}\
=&\lim_{x
ightarrow - 24}\frac{(x^{2}+49)-625}{(x + 24)(\sqrt{x^{2}+49}+25)}\
=&\lim_{x
ightarrow - 24}\frac{x^{2}-576}{(x + 24)(\sqrt{x^{2}+49}+25)}
\end{align*}
$$
## Step2: Factor the numerator
Factor $x^{2}-576=(x + 24)(x - 24)$.
$$
\begin{align*}
&\lim_{x
ightarrow - 24}\frac{(x + 24)(x - 24)}{(x + 24)(\sqrt{x^{2}+49}+25)}\
=&\lim_{x
ightarrow - 24}\frac{x - 24}{\sqrt{x^{2}+49}+25}
\end{align*}
$$
## Step3: Substitute $x=-24$
$$
\begin{align*}
&\frac{-24-24}{\sqrt{(-24)^{2}+49}+25}\
=&\frac{-48}{\sqrt{576 + 49}+25}\
=&\frac{-48}{\sqrt{625}+25}\
=&\frac{-48}{25 + 25}\
=&-\frac{24}{25}
\end{align*}
$$
# Answer:
$-\frac{24}{25}$

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QUESTION IMAGE

Question

Explanation:

Step1: Rationalize the numerator

Step2: Factor the numerator

Step3: Substitute $x=-24$

Answer: