Question

Please Help Quickly!!! Find the limit if f(x) = x^3

Answer 1

Answer:

Option b. 12

Step-by-step explanation:

This exercise asks us to find the derivative of a function using the definition of a derivative.

Our function is $f(x) = x^{3}$. Therefore:

$f(2+h) = (2+h)^{3}$

$f(2) = (2)^{3} = 8$

Then:

$\lim_{h \to \0} \frac{f(2+h)-f(2)}{h}=\lim_{h \to \0} \frac{(2+h)^{3}-8}{h}$

Expanding:

$\lim_{h \to \0} \frac{(2+h)^{3}-8}{h} =\lim_{h \to \0} \frac{8+ h^{3} +6h(2+h) -8}{h} =\lim_{h \to \0} \frac{h^{3} +6h(2+h)}{h}$

$\lim_{h \to \0} \frac{h^{3}+ 6h(2+h)}{h} =\lim_{h \to \0} h^{2} + 6(2+h)$

Now, if x=0:

$\lim_{h \to \0} \frac{f(2+h)-f(2)}{h} = (0)^{2} +6(2+0) = 12$