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Question Number 106869 by bemath last updated on 07/Aug/20

$$@\mathrm{bemath}@$$$$\underset{x\to \pi }{\lim }\left[\left(\frac{1}{2\mathrm{x}-2\pi }\right)\underset{\pi }{\stackrel{\mathrm{x}}{\int }}\frac{\cos 2\mathrm{t}\mathrm{dt}}{1-\cos 3\mathrm{t}}\right]=?$$

Answered by bobhans last updated on 07/Aug/20

$$\stackrel{\wedge \mathrm{bobhans}\wedge }{}$$$$\underset{x\to \pi }{\lim }\left[\frac{\underset{\pi }{\stackrel{\mathrm{x}}{\int }}\frac{\cos 2\mathrm{t}\mathrm{dt}}{1-\cos 3\mathrm{t}}}{2\mathrm{x}-2\pi }\right]=\underset{x\to \pi }{\lim }\frac{\frac{\mathrm{d}}{\mathrm{dx}}\stackrel{\mathrm{x}}{\int }\frac{\cos 2\mathrm{t}\mathrm{dt}}{1-\cos 3\mathrm{t}}}{\frac{\mathrm{d}}{\mathrm{dx}}[2\mathrm{x}-2\pi ]}$$$$=\underset{x\to \pi }{\lim }\frac{\left(\frac{\cos 2\mathrm{x}}{1-\cos 3\mathrm{x}}\right)}{2}=\frac{\frac{1}{1+1}}{2}=\frac{1}{4}$$

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