Question

use properties of limits to find the indicated limit. it may be necessary to rewrite the expression before limit properties can be applied.
lim_{x
ightarrow5}\frac{x^{2}+2x - 35}{x^{2}-25}
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. lim_{x
ightarrow5}\frac{x^{2}+2x - 35}{x^{2}-25}=\text{ (type an integer or a simplified fraction.)}
b. the limit does not exist and is neither (infty) nor (-infty).

Explanation:

Step1: Factor the numerator and denominator

The numerator $x^{2}+2x - 35=(x + 7)(x-5)$ and the denominator $x^{2}-25=(x + 5)(x - 5)$. So the limit becomes $\lim_{x
ightarrow5}\frac{(x + 7)(x - 5)}{(x + 5)(x - 5)}$.

Step2: Cancel out the common factor

Since $x
eq5$ when taking the limit as $x
ightarrow5$, we can cancel out the common factor $(x - 5)$ in the numerator and denominator. The expression simplifies to $\lim_{x
ightarrow5}\frac{x + 7}{x + 5}$.

Step3: Substitute the value of $x$

Substitute $x = 5$ into $\frac{x+7}{x + 5}$. We get $\frac{5+7}{5+5}=\frac{12}{10}=\frac{6}{5}$.

Answer:

A. $\lim_{x
ightarrow5}\frac{x^{2}+2x - 35}{x^{2}-25}=\frac{6}{5}$