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Question Number 187270 by MikeH last updated on 15/Feb/23

$$\underset{x\to \pi }{\lim }\frac{\cos x+\sin \left(2x\right)+1}{x^2-{\pi }^2}=$$$$\mathrm{A}.\frac{1}{2\pi }C.1$$$$B.\frac{1}{\pi }D.\mathrm{Does}\mathrm{Not}\mathrm{exist}$$

Answered by horsebrand11 last updated on 15/Feb/23

$$\underset{x\to \pi }{\lim }\frac{1-\cos (\pi -x)-\sin 2(\pi -x)}{(x+\pi )(x-\pi )}$$$$=\frac{1}{2\pi }.\underset{x\to 0}{\lim }\frac{1-\cos x-\sin 2x}{-x}$$$$=-\frac{1}{2\pi }[\underset{x\to 0}{\lim }\frac{1-\cos x}{x}-\underset{x\to 0}{\lim }\frac{\sin 2x}{x}]$$$$=-\frac{1}{2\pi }[0-2]=\frac{1}{\pi }$$

Commented by MikeH last updated on 16/Feb/23

$$\mathrm{thanks}$$

Answered by Frix last updated on 15/Feb/23

$$\mathrm{Simply}\mathrm{use}{\mathrm{l}}^{\prime }\mathrm{H}\widehat{\mathrm{o}pital}\Rightarrow \mathrm{answer}\mathrm{is}\frac{1}{\pi }$$

Answered by CElcedricjunior last updated on 15/Feb/23

$$\underset{x\to \mathit{\pi }}{\lim }\frac{\mathit{c}\mathit{o}\mathit{s}\mathit{x}+\mathit{s}\mathit{i}\mathit{n}\left(2\mathit{x}\right)+1}{{\mathit{x}}^2-{\mathit{\pi }}^2}=\frac{0}{0}=\mathit{F}\mathbf{I}◼\mathit{M}oi\mathit{v}\mathit{r}\mathit{e}$$$$\underset{x\to \mathit{\pi }}{\lim }\frac{-\mathit{s}\mathit{i}\mathit{n}\mathit{x}+2\mathit{c}\mathit{o}\mathit{s}\left(2\mathit{x}\right)}{2\mathit{x}}=\frac{2}{2\mathit{\pi }}=\frac{1}{\mathit{\pi }}★cedricjunior$$$$\mathit{B})\frac{1}{\mathit{\pi }}$$

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