What is the limit as h approaches 0 of ((x+h)^{2/3}-(x^{2/3}))/h ?

The limit as h approaches 0 of ((x+h)^{2/3}-(x^{2/3}))/h is 2/(3x^{1/3)}

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limit as h approaches 0 of ((x+h)^{2/3}-(x^{2/3}))/h

h \to 0 lim \frac{( x + h ) ^{\frac{2}{3}} - ( x ^{\frac{2}{3}} )}{h}

\frac{2}{3 x ^{\frac{1}{3}}}

Solution steps

h \to 0 lim (\frac{( x + h ) ^{\frac{2}{3}} - x ^{\frac{2}{3}}}{h})

Apply L'Hopital's Rule

= h \to 0 lim \frac{\frac{2}{3 ( h + x ) ^{\frac{1}{3}}}}{1}

Plug in the value

h = 0

= \frac{\frac{2}{3 ( 0 + x ) ^{\frac{1}{3}}}}{1}

Simplify

= \frac{2}{3 x ^{\frac{1}{3}}}

What is the limit as h approaches 0 of ((x+h)^{2/3}-(x^{2/3}))/h ?
The limit as h approaches 0 of ((x+h)^{2/3}-(x^{2/3}))/h is 2/(3x^{1/3)}