Lim as h -->0 ((x+h)^(3) - x^(3))/h

lim as h -->0 ((x+h)^(3) - x^(3))/h

step by step please.

thanks

you have to do the tedious work. first expand (x+h)^3
if you know Pascal's triangle, you can write it out immediately (the part in square brackets is the expansion of x plus h cubed):

{[x^3 + 3x^2h + 3xh^2 + h^3] - x^3 } / h
this simplifies to
3x^2 + 3xh + h^2

now when h->0, this becomes nothing more than 3x^2 since the second and third terms go to zero and we're left with just the first term.

Take Angie's advice, expand

Lim h -->0 ((x+h)³ - x³)/h = Lim[h ->0](x³ + 3x²h + 3xh² + h³ - x³)/h

= Lim[h ->0](3x²h + 3xh² + h³)/h = Lim[h ->0](3x² + 3xh + h²)

= 3x² + 0 + 0 = 3x²

I bet you thought Calculus did not require Algebra, surprise!!!

ProfRay

1st ; YOU compute [ x + h ]^3
2nd ;YOU subtract x^3 from your answer to '1st'
3rd ; divide your answer to 2nd by h
4th; let h tend to zero to find " 3x ² "....nothing more than competency in algebra

lim(h→0) [ (x + h)³ – x³ ] / h =
lim(h→0) [ x³ + 3x²h + 3xh² + h³ – x³ ] / h =
lim(h→0) [ 3x²h + 3xh² + h³ ] / h =
lim(h→0) 3x² + 3xh + h² =
∙ ∙ ∙ ∙ ∙ ∙ ∙ 3x² + 3x(0) + 0 = 3x²