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  4. displaystyle lim x arrow 0 (x √[3]z2-(z-x)2/(√[3]8 x z-4 x2+√[3]8 x z)4) is equal to

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Q. $\displaystyle\lim _{x \rightarrow 0} \frac{x \sqrt[3]{z^{2}-(z-x)^{2}}}{\left(\sqrt[3]{8 x z-4 x^{2}}+\sqrt[3]{8 x z}\right)^{4}}$ is equal to

Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow 0} \frac{x \sqrt[3]{z^{2}-(z-x)^{2}}}{\left(\sqrt[3]{8 x z-4 x^{2}}+\sqrt[3]{8 x z}\right)^{4}}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{x \sqrt[3]{2 x z-x^{2}}}{(\sqrt[3]{x} \sqrt[3]{8 z-4 x}+\sqrt[3]{8 z} \sqrt[3]{x})^{4}}$
$=\displaystyle\lim _{x \rightarrow 0} \frac{x^{4 / 3} \sqrt[3]{2 z-x}}{x^{4 / 3}[\sqrt[3]{8 z-4 x}+\sqrt[3]{8 z}]^{4}}$
$=\frac{\sqrt[3]{2 z}}{[2 \sqrt[3]{8 z}]^{4}}=\frac{1}{2^{23 / 3} \cdot z}$