use properties of limits to find the indicated limit. it may be necessary to rewrite an expression before…

use properties of limits to find the indicated limit. it may be necessary to rewrite an expression before limit properties can be applied. lim(x→3) x/(x + 5) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim(x→3) x/(x + 5) = (simplify your answer. type an integer or a simplified fraction.) b. the limit does not exist.

use properties of limits to find the indicated limit. it may be necessary to rewrite an expression before limit properties can be applied. lim(x→3) x/(x + 5) select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. lim(x→3) x/(x + 5) = (simplify your answer. type an integer or a simplified fraction.) b. the limit does not exist.

Answer

Explanation:

Step1: Apply limit - quotient rule

If $\lim_{x\rightarrow a}f(x)$ and $\lim_{x\rightarrow a}g(x)$ exist and $\lim_{x\rightarrow a}g(x)\neq0$, then $\lim_{x\rightarrow a}\frac{f(x)}{g(x)}=\frac{\lim_{x\rightarrow a}f(x)}{\lim_{x\rightarrow a}g(x)}$. Here $f(x) = x$ and $g(x)=x + 5$, and $a = 3$.

Step2: Find $\lim_{x\rightarrow3}x$ and $\lim_{x\rightarrow3}(x + 5)$

By the direct - substitution property of limits, $\lim_{x\rightarrow3}x=3$ and $\lim_{x\rightarrow3}(x + 5)=\lim_{x\rightarrow3}x+\lim_{x\rightarrow3}5=3 + 5=8$.

Step3: Calculate the limit of the quotient

$\lim_{x\rightarrow3}\frac{x}{x + 5}=\frac{\lim_{x\rightarrow3}x}{\lim_{x\rightarrow3}(x + 5)}=\frac{3}{8}$.

Answer:

A. $\lim_{x\rightarrow3}\frac{x}{x + 5}=\frac{3}{8}$