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  4. displaystyle lim x arrow-1((x4+x2+x+1/x2-x+1))(1- cos (x+1)/(x+1)2) is equal to

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Q. $\displaystyle\lim _{x \rightarrow-1}\left(\frac{x^{4}+x^{2}+x+1}{x^{2}-x+1}\right)^{\frac{1-\cos (x+1)}{(x+1)^{2}}}$ is equal to

Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow-1}\left(\frac{x^{4}+x^{2}+x+1}{x^{2}-x+1}\right)^{\frac{1-\cos (x+1)}{(x+1)^{2}}}$
$=\displaystyle\lim _{x \rightarrow-1}\left(\frac{x^{4}+x^{2}+x+1}{x^{2}-x+1}\right)^{\frac{2 \sin ^{2}\left(\frac{x+1}{2}\right)}{(x+1)^{2}}}$
$=\displaystyle\lim _{x \rightarrow-1}\left(\frac{x^{4}+x^{2}+x+1}{x^{2}-x+1}\right)^{\frac{1}{2}}\left(\frac{\sin \left(\frac{x+1}{2}\right)}{\left(\frac{x+1}{2}\right)}\right)^{2}=\left(\frac{2}{3}\right)^{1 / 2}$