Question

given the function $f(x)=3ln(x + 2)$, determine the instantaneous rate of change of $f$ at $x = 5$ using the limit shown below. you do not have to simplify your answer.
answer attempt 1 out of 2
$lim_{h
ightarrow0}$

Explanation:

Step1: Recall the formula for instantaneous rate of change

The instantaneous rate of change of a function $y = f(x)$ at $x=a$ is given by $\lim_{h
ightarrow0}\frac{f(a + h)-f(a)}{h}$. Here, $a = 5$ and $f(x)=3\ln(x + 2)$.

Step2: Find $f(5 + h)$ and $f(5)$

First, $f(5+h)=3\ln((5 + h)+ 2)=3\ln(h + 7)$. Second, $f(5)=3\ln(5 + 2)=3\ln(7)$.

Step3: Substitute into the limit formula

The limit is $\lim_{h
ightarrow0}\frac{3\ln(h + 7)-3\ln(7)}{h}$.

Answer:

$\lim_{h
ightarrow0}\frac{3\ln(h + 7)-3\ln(7)}{h}$