Question

1. - / 3.84 points 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}$

Explanation:

Step1: Rationalize the numerator

Multiply the fraction by $\frac{\sqrt{x^{2}+49}+25}{\sqrt{x^{2}+49}+25}$. We get $\lim_{x
ightarrow - 24}\frac{(\sqrt{x^{2}+49}-25)(\sqrt{x^{2}+49}+25)}{(x + 24)(\sqrt{x^{2}+49}+25)}$. Using the difference - of - squares formula $(a - b)(a + b)=a^{2}-b^{2}$, the numerator becomes $x^{2}+49 - 625=x^{2}-576$.

Step2: Factor the numerator

Factor $x^{2}-576$ as $(x + 24)(x - 24)$. So the limit is $\lim_{x
ightarrow - 24}\frac{(x + 24)(x - 24)}{(x + 24)(\sqrt{x^{2}+49}+25)}$.

Step3: Cancel out the common factor

Cancel out the common factor $(x + 24)$ (since $x
eq - 24$ when taking the limit), we have $\lim_{x
ightarrow - 24}\frac{x - 24}{\sqrt{x^{2}+49}+25}$.

Step4: Substitute $x=-24$

Substitute $x = - 24$ into $\frac{x - 24}{\sqrt{x^{2}+49}+25}$. We get $\frac{-24-24}{\sqrt{(-24)^{2}+49}+25}=\frac{-48}{\sqrt{576 + 49}+25}=\frac{-48}{\sqrt{625}+25}=\frac{-48}{25 + 25}=-\frac{24}{25}$.

Answer:

$-\frac{24}{25}$