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→0 (1/(x + 8) - 1/8)/x select the correct choice below and fill in any answer boxes in your choice. o a. lim x→0 (1/(x + 8) - 1/8)/x = (type an integer or a simplified fraction.) o 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→0 (1/(x + 8) - 1/8)/x select the correct choice below and fill in any answer boxes in your choice. o a. lim x→0 (1/(x + 8) - 1/8)/x = (type an integer or a simplified fraction.) o b. the limit does not exist.

Answer

Explanation:

Step1: Combine the fractions in the numerator

First, find a common - denominator for $\frac{1}{x + 8}-\frac{1}{8}$. The common denominator is $8(x + 8)$. So, $\frac{1}{x + 8}-\frac{1}{8}=\frac{8-(x + 8)}{8(x + 8)}=\frac{8 - x - 8}{8(x + 8)}=\frac{-x}{8(x + 8)}$. Then the original limit $\lim_{x\rightarrow0}\frac{\frac{1}{x + 8}-\frac{1}{8}}{x}$ becomes $\lim_{x\rightarrow0}\frac{\frac{-x}{8(x + 8)}}{x}$.

Step2: Simplify the complex - fraction

$\frac{\frac{-x}{8(x + 8)}}{x}=\frac{-x}{8(x + 8)}\cdot\frac{1}{x}=\frac{-1}{8(x + 8)}$ (for $x\neq0$).

Step3: Apply the limit property

Now, find $\lim_{x\rightarrow0}\frac{-1}{8(x + 8)}$. Substitute $x = 0$ into the expression $\frac{-1}{8(x + 8)}$. We get $\frac{-1}{8(0 + 8)}=-\frac{1}{64}$.

Answer:

$-\frac{1}{64}$