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  4. displaystyle lim x arrow 1 (√[3]x2-2 √[3]x+1/(x-1)2) is equal to

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Q. $\displaystyle\lim _{x \rightarrow 1} \frac{\sqrt[3]{x^{2}}-2 \sqrt[3]{x}+1}{(x-1)^{2}}$ is equal to

Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow 1} \frac{\sqrt[3]{x^{2}}-2 \sqrt[3]{x}+1}{(x-1)^{2}}$
$=\displaystyle\lim _{y \rightarrow 1} \frac{y^{2}-2 y+1}{\left(y^{3}-1\right)^{2}}$
$[$ Putting $\sqrt[3]{x}=y ;$ as $x \rightarrow 1, y \rightarrow 1]$
$=\displaystyle\lim _{y \rightarrow 1} \frac{(y-1)^{2}}{(y-1)^{2}\left(y^{2}+y+1\right)^{2}}$
$=\displaystyle\lim _{y \rightarrow 1} \frac{1}{\left(y^{2}+y+1\right)^{2}}=\frac{1}{9}$