Question

Lim h=>0 [(x+h) ^1/3-x^1/3] /h

Answer 1

$\underset{h\to0}{\lim}\ \dfrac{(x+h)^\frac13-x^\frac13}{h}=\\\\\\\underset{h\to0}{\lim}\ \dfrac{(x+h)^\frac13-x^\frac13}{h}\times\dfrac{(x+h)^\frac23+(x+h)^\frac13x^\frac13+x^\frac23}{(x+h)^\frac23+(x+h)^\frac13x^\frac13+x^\frac23}=\\\\\\\underset{h\to0}{\lim}\ \dfrac{(x+h)-x}{h((x+h)^\frac23+(x+h)^\frac13x^\frac13+x^\frac23)}=\\\\\\\underset{h\to0}{\lim}\ \dfrac{1}{((x+h)^\frac23+(x+h)^\frac13x^\frac13+x^\frac23)}=\\\\\\\dfrac{1}{((x+0)^\frac23+(x+0)^\frac13x^\frac13+x^\frac23)}=$

$\dfrac{1}{2x^\frac23+x^\frac23}=\boxed{\dfrac{1}{3x^\frac23}}$