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

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

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

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

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

Solution steps

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

Apply L'Hopital's Rule

: h \to 0 lim \frac{\frac{2}{3 ( h + a ) ^{\frac{1}{3}}}}{1}

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

Plug in the value

h = 0

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

Simplify

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

x \to 2 + lim (2 x^{2} + \frac{a}{2} x - 4)

x \to 5 lim (\frac{x ^{2} - 4}{x - 2})

x \to 2 lim (\frac{4}{x ^{2} - 4})

x \to \infty lim (\frac{2}{3 + 4 ^{\frac{1}{x}}})

x \to \infty lim (- 7 x - 4)

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