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\(\displaystyle{ \lim_{x \to \pi+} \frac{xsinx}{cosx+1}}\)

\(\displaystyle{ \lim_{x \to \pi+} \frac{x \sin x}{\cos x+1}= \lim_{x \to \pi+} \frac{xsinx}{2 \cos^2 \frac{x}{2}}=\lim_{x \to \pi+} \frac{x*2 \sin \frac{x}{2} \cos \frac{x}{2} }{2 \cos^2 \frac{x}{2}}=\lim_{x \to \pi+} \frac{x \sin \frac{x}{2}}{\cos \frac{x}{2}}=\lim_{x \to \pi+} x \tg \frac{x}{2}= \pi (-\infty)=-\infty}\)

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