Given f(x) = x3, evaluate lim h->o f(a+h)-f(a)/h

Answer 1

f(a+h)= (a+h)^3= (a^2+ah+h^2)(a+h)=a^3+ha^2+h^2a+ha^2+ah^2+h^3

f(a)= a^3

so f(a+h)-f(a)= a^3+ha^2+h^2a+ha^2+ah^2+h^3-a^3
= ha^2+h^2a+ha^2+ah^2+h^3

dividing that by h gives
= a^2 + ah+ a^2 +ah^2 + h^2

= 2a^2 as the h approaches zero

Answer 2

I respectfully have to disagree with bob

(x+h)^3 = a^3 + 3a^2h + 3ah^2 + h^3

which when inserted in the above calculation results in a final of

3a^2