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Question Number 157314 by cortano last updated on 22/Oct/21

$$\underset{x\to \mathrm{\infty }}{\lim }\frac{3\ln (1+5\tan \frac{4}{x})}{x(1-\cos \frac{6}{x})}=?$$

Commented by john_santu last updated on 22/Oct/21

$$let\frac{1}{x}=uandu\to 0$$$$\underset{u\to 0}{\lim }\frac{3u\ln (1+5\tan 4u)}{1-\cos 6u}$$$$=\underset{u\to 0}{\lim }\frac{3u\ln (1+5\tan 4u)}{2\sin {}^{2}3u}$$$$=\underset{u\to 0}{\lim }\left(\frac{3u}{2\sin 3u}\right).\underset{u\to 0}{\lim }\left(\frac{\ln (1+5\tan 4u)}{\sin 3u}\right)$$$$=\frac{1}{2}\times \underset{u\to 0}{\lim }\frac{\ln (1+5\tan 4u)}{5\tan 4u}\times \frac{5\tan 4u}{\sin 3u}$$$$=\frac{1}{2}\times 1\times \frac{20}{3}=\frac{10}{3}$$

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