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  4. If L = displaystyle lim x arrow 2 ((10- x )1 / 3-2/ x -2), then the value of |1 /(4 L )| is

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Q. If $L =\displaystyle\lim _{ x \rightarrow 2} \frac{(10- x )^{1 / 3}-2}{ x -2}$, then the value of $|1 /(4 L )|$ is

Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow 2} \frac{(10-x)^{1 / 3}-2}{x-2}$
$=\displaystyle\lim _{h \rightarrow 0} \frac{(8-h)^{1 / 3}-2}{h}$
$($ Put $x=2+h)$
$=\displaystyle\lim _{h \rightarrow 0} \frac{2\left(1-\frac{h}{8}\right)^{1 / 3}-2}{h}$
$=2 \displaystyle\lim _{h \rightarrow 0} \frac{\left(1-\frac{h}{8}\right)^{1 / 3}-1}{h}$
$=2 \displaystyle\lim _{h \rightarrow 0} \frac{1-\frac{1}{3} \frac{h}{8}-1}{h}$
$=-\frac{1}{12}$