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  4. The value of displaystyle lim x arrow 1 (√[3](7-x)-2/(x+1)) is

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Q. The value of $\displaystyle\lim _{x \rightarrow 1} \frac{\sqrt[3]{(7-x)}-2}{(x+1)}$ is

Limits and Derivatives

Solution:

Let $x+1=h$
Then $\displaystyle\lim _{x \rightarrow-1} \frac{\sqrt[3]{(7-x)}-2}{(x+1)}$
$=\displaystyle\lim _{h \rightarrow 0} \frac{(8-h)^{1 / 3}-2}{h}$
$=\displaystyle\lim _{h \rightarrow 0} \frac{2\left(1-\frac{h}{8}\right)^{1 / 3}-2}{h}$
$=2\displaystyle \lim _{h \rightarrow 0} \frac{\left(1-\frac{1}{3} \frac{h}{8}\right)-1}{h}=-\frac{1}{12}$