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  4. displaystyle lim x arrow 1 (1+ sin π((3 x/1+x2))/1+ cos π x) is equal to

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Q. $\displaystyle\lim _{x \rightarrow 1} \frac{1+\sin \pi\left(\frac{3 x}{1+x^{2}}\right)}{1+\cos \pi x}$ is equal to

Limits and Derivatives

Solution:

$\displaystyle\lim _{x \rightarrow 1} \frac{1+\sin \pi\left(\frac{3 x}{1+x^{2}}\right)}{1+\cos \pi x}$
$=\displaystyle\lim _{x \rightarrow 1} \frac{1-\cos \left(\frac{3 \pi}{2}-\frac{3 \pi x}{1+x^{2}}\right)}{1-\cos (\pi-\pi x)}$
$=\displaystyle\lim _{x \rightarrow 1} \frac{2 \sin ^{2}\left(\frac{3 \pi}{4}-\frac{3 \pi x}{2\left(1+x^{2}\right)}\right)}{2 \sin ^{2}\left(\frac{\pi}{2}-\frac{\pi x}{2}\right)}$
$=\displaystyle\lim _{x \rightarrow 1}\left(\frac{\frac{3 \pi}{4}-\frac{3 \pi x}{2\left(1+x^{2}\right)}}{\frac{\pi}{2}-\frac{\pi x}{2}}\right)^{2}$
$=\displaystyle\lim _{x \rightarrow 1} 9\left(\frac{\frac{1}{2}-\frac{x}{1+x^{2}}}{1-x}\right)^{2}$
$=\displaystyle\lim _{x \rightarrow 1} 9\left(\frac{x-1}{2\left(1+x^{2}\right)}\right)^{2}=0$