lim_(h -> 0) ((x+h)^(3)-x^(3))/(h) Recall that: A^(3)-B^(3)=(A-B)(A^(2)+AB+B^(2)) =lim_(h -> 0) ((x+h-x)[(x+h)^(2)+(x+h)x+x^(2)])/(h) =lim_((h)[(x+h)^(2)+(x+h)x+x^(2)])/(h) Cancel h from both the numerator and the denominator =lim_(h -> 0) (x+h)^(2)+(x+h)x+x^(2) Note that the denominator is no longer zero upon direct substitution =(x+0)^(2)+(x+0)x+x^(2)=x^(2)+x^(2)+x^(2)=3x^(2)

Let's calculate the limit of the numerator and the limit of the denominator. (0)/(0) Insofar as (0)/(0) is uncertainty, L'Hôpital's rule applies. L'Hôpital's rule states that the limit of the quotient of functions is equal to the limit of their partial derivatives. lim_(h -> 0) ((x+h)^(3)-x^(3))/(h)=lim_(h -> 0) ((d)/(dh)[(x+h)^(3)-x^(3)])/((d)/(dh)[h]) Find the Derivative of the Numerator and Denominator lim_(h -> 0) (3(x+h)^(2))/(1) Let's take the limit of each term. (3(lim_(h -> 0) x+ lim_(h -> 0) h)^(2))/(lim1_(h -> 0)) Determine limits by substituting 0 for all occurrences h. (3(x+0)^(2))/(1) 3x^(2)