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  4. The value of displaystyle limx→2 ((x-2)/(x3-x2-x-2)) is equal to

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Q. The value of $ \displaystyle\lim_{x\to2} \frac{\left(x-2\right)}{\left(x^{3}-x^{2}-x-2\right)} $ is equal to

J & K CETJ & K CET 2016Limits and Derivatives

Solution:

We have, $\lim\limits_{x\to2}\left(\frac{x-2}{x^{3}-x^{2}-x-2}\right)$
$=\lim\limits_{x\to2}\left(\frac{x-2}{\left(x-2\right)\left(x^{2}+x+1\right)}\right) $
$=\lim\limits_{x\to2}\left(\frac{1}{x^{2}+x+1}\right) $
$= \frac{1}{4+2+1} = \frac{1}{7}$