Question

question 6
0/1 pt 5 19 details
the limit below represents the derivative of some function $f(x)$ at some number $a$.
$\lim_{h \to 0} \frac{5(2+h)^4 - 80}{h}$
state the function and the number:
$f(x) = $
$a = $
question help: video

Explanation:

Step1: Recall derivative definition

The derivative of $f(x)$ at $x=a$ is given by:
$$f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$$

Step2: Match to given limit

Compare with $\lim_{h \to 0} \frac{5(2+h)^4 - 80}{h}$. Identify $a+h=2+h$, so $a=2$. Then $f(a)=f(2)=80$, and $f(a+h)=5(a+h)^4$, so $f(x)=5x^4$. Verify: $f(2)=5(2)^4=5*16=80$, which matches.

Answer:

$f(x) = 5x^4$
$a = 2$