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Question Number 24864 by A1B1C1D1 last updated on 27/Nov/17

$$\mathrm{Solve}\mathrm{the}\mathrm{following}\mathrm{trigonometric}\mathrm{limit}:$$$$$$$$\underset{\mathrm{x}\to \frac{\pi }{4}}{\lim }\left(5\mathrm{tg}\left(\mathrm{x}\right)\right)=$$

Answered by jota+ last updated on 28/Nov/17

$$\underset{x\to 0}{\lim }sinx=0\underset{x\to 0}{\lim }cosx=1$$$$\Rightarrow \underset{x\to 0}{\lim tan}x=0$$$$h=x-\frac{\pi }{4}$$$$5\tan x=$$$$=5\tan (\frac{\pi }{4}+\mathrm{h})=5\frac{1+\tan h}{1-\tan h}.$$$$\underset{x\to \pi /4}{\lim 5tan}x=5\underset{h\to 0}{\lim }\frac{1+\tan h}{1-\tan h}=5$$

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