use the properties of limits to help decide whether the limit exists. if the limit exists, find its value.\n…

use the properties of limits to help decide whether the limit exists. if the limit exists, find its value.\n lim_{h\rightarrow0}\frac{\frac{9}{7 + h}-\frac{9}{7}}{h} \nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. lim_{h\rightarrow0}\frac{\frac{9}{7 + h}-\frac{9}{7}}{h}=-\frac{9}{49} \nb. the limit does not exist and is neither (infty) nor (-infty).

use the properties of limits to help decide whether the limit exists. if the limit exists, find its value.\n lim_{h\rightarrow0}\frac{\frac{9}{7 + h}-\frac{9}{7}}{h} \nselect the correct choice below and, if necessary, fill in the answer box to complete your choice.\na. lim_{h\rightarrow0}\frac{\frac{9}{7 + h}-\frac{9}{7}}{h}=-\frac{9}{49} \nb. the limit does not exist and is neither (infty) nor (-infty).

Answer

Explanation:

Step1: Combine the fractions in the numerator

First, find a common - denominator for $\frac{9}{7 + h}-\frac{9}{7}$. The common denominator is $7(7 + h)$. So, $\frac{9}{7 + h}-\frac{9}{7}=\frac{9\times7-9\times(7 + h)}{7(7 + h)}=\frac{63-63 - 9h}{7(7 + h)}=\frac{-9h}{7(7 + h)}$.

Step2: Substitute into the original limit

The original limit $\lim_{h\rightarrow0}\frac{\frac{9}{7 + h}-\frac{9}{7}}{h}$ becomes $\lim_{h\rightarrow0}\frac{\frac{-9h}{7(7 + h)}}{h}=\lim_{h\rightarrow0}\frac{-9h}{7(7 + h)\times h}$.

Step3: Simplify the expression

Cancel out the $h$ terms in the numerator and denominator. We get $\lim_{h\rightarrow0}\frac{-9}{7(7 + h)}$.

Step4: Evaluate the limit

Substitute $h = 0$ into the expression $\frac{-9}{7(7 + h)}$. We have $\frac{-9}{7\times(7+0)}=-\frac{9}{49}$.

Answer:

A. $\lim_{h\rightarrow0}\frac{\frac{9}{7 + h}-\frac{9}{7}}{h}=-\frac{9}{49}$