Answer:
The value of the limit is 12.
Step-by-step explanation:
Small typing mistake, the limit is of h tending to 0.
We have that:
[tex]f(x) = x^{3}[/tex]
Then
[tex]f(2+h) = (2+h)^{3} = 8 + 12h + 6h^{2} + h^{3}[/tex]
[tex]f(2) = 2^{3} = 8[/tex]
Calling the limit L
[tex]L = \frac{f(2+h) - f(2)}{h}[/tex]
[tex]L = \frac{8 + 12h + 6h^{2} + h^{3} - 8}{h}[/tex]
[tex]L = \frac{h^{3} + 6h^{2} + 12h}{h}[/tex]
h is the common term in the numerator, then
[tex]L = \frac{h(h^{2} + 6h + 12)}{h}[/tex]
Simplifying by h
[tex]L = h^{2} + 6h + 12[/tex]
Since h tends to 0.
[tex]L = 0^{2} + 6*0 + 12[/tex]
[tex]L = 12[/tex]
So the answer is 12.
