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Question Number 142459 by bramlexs22 last updated on 01/Jun/21

$$-----------$$$$\underset{x\to \frac{\pi }{2}}{\lim }(\frac{x}{\cot x}-\frac{\pi }{2\cos x})=?$$$$\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}\mathrm{\_}$$

Answered by benjo_mathlover last updated on 01/Jun/21

$$\underset{x\to \frac{\pi }{2}}{\lim }(\frac{x\sin x}{\cos x}-\frac{\pi }{2\cos x})$$$$=\underset{x\to \frac{\pi }{2}}{\lim }\left(\frac{2x\sin x-\pi }{2\cos x}\right)$$$$=\underset{x\to \frac{\pi }{2}}{\lim }\left(\frac{2\sin x+2x\cos x}{-2\sin x}\right)$$$$=-1$$

Answered by mathmax by abdo last updated on 02/Jun/21

$$\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{x}}{\mathrm{cotanx}}-\frac{\pi }{2\mathrm{cosx}}\Rightarrow \mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{xsinx}}{\mathrm{cosx}}-\frac{\pi }{2\mathrm{cosx}}=\frac{2\mathrm{xsinx}-\pi }{2\mathrm{cosx}}$$$$\mathrm{changement}\mathrm{x}=\frac{\pi }{2}-\mathrm{t}\mathrm{give}\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}(\frac{\pi }{2}-\mathrm{t})=\frac{2(\frac{\pi }{2}-\mathrm{t})\mathrm{cost}-\pi }{2\mathrm{sint}}$$$$=\frac{(\pi -2\mathrm{t})\mathrm{cost}-\pi }{2\mathrm{sint}}\sim \frac{(\pi -2\mathrm{t})(1-\frac{{\mathrm{t}}^2}{2})-\pi }{2\mathrm{t}}$$$$=\frac{\pi -\frac{\pi }{2}{\mathrm{t}}^2-2\mathrm{t}+{\mathrm{t}}^3-\pi }{2\mathrm{t}}=-\frac{\pi }{4}\mathrm{t}-1+\frac{{\mathrm{t}}^2}{2}\to -1(\mathrm{t}\to 0)\Rightarrow$$$${\lim }_{\mathrm{x}\to \frac{\pi }{2}}\mathrm{f}\left(\mathrm{x}\right)=-1$$

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