What is the limit as x approaches a of (\sqrt[3]{x}-\sqrt[3]{a})/(x^2-a^2) ?

The limit as x approaches a of (\sqrt[3]{x}-\sqrt[3]{a})/(x^2-a^2) is 1/(6a^{5/3)}

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limit as x approaches a of (\sqrt[3]{x}-\sqrt[3]{a})/(x^2-a^2)

\frac{1}{6 a ^{\frac{5}{3}}}

Solution steps

Apply L'Hopital's Rule

= x \to a lim (\frac{\frac{1}{3 x ^{\frac{2}{3}}}}{2 x})

Plug in the value

x = a

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

Simplify

\frac{\frac{1}{3 a ^{\frac{2}{3}}}}{2 a} : \frac{1}{6 a ^{\frac{5}{3}}}

= \frac{1}{6 a ^{\frac{5}{3}}}

What is the limit as x approaches a of (\sqrt[3]{x}-\sqrt[3]{a})/(x^2-a^2) ?
The limit as x approaches a of (\sqrt[3]{x}-\sqrt[3]{a})/(x^2-a^2) is 1/(6a^{5/3)}