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Q. $\displaystyle\lim _{h \rightarrow 0}\left(\frac{1}{h \sqrt[3]{8+h}}-\frac{1}{2 h}\right)$ equals to

Limits and Derivatives

Solution:

$\displaystyle\lim _{h \rightarrow 0} \frac{2-\sqrt[3]{8+h}}{2 h \cdot \sqrt[3]{8+h}}$
$=\displaystyle\lim _{h \rightarrow 0} \frac{8-(8+h)}{2 h \cdot \sqrt[3]{8+h}\left\{8^{2 / 3}+8^{1 / 3} \cdot(8+h)^{1 / 3}+(8+h)^{2 / 3}\right\}}$
$=-\frac{1}{48}$