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Question Number 21223 by moadakennaf last updated on 16/Sep/17

lim_(x→π/2) (π−2x)tan (x)

$$\underset{{x}\rightarrow\pi/\mathrm{2}} {\mathrm{lim}}\left(\pi−\mathrm{2}{x}\right)\mathrm{tan}\:\left({x}\right) \\ $$

Answered by dioph last updated on 16/Sep/17

= lim_(x→π/2)  ((π−2x)/(cot x)) =  = lim_(x→π/2)  ((−2)/(−cosec^2  x)) =  = lim_(x→π/2)  2 sin^2  x =  = 2

$$=\:\underset{{x}\rightarrow\pi/\mathrm{2}} {\mathrm{lim}}\:\frac{\pi−\mathrm{2}{x}}{\mathrm{cot}\:{x}}\:= \\ $$$$=\:\underset{{x}\rightarrow\pi/\mathrm{2}} {\mathrm{lim}}\:\frac{−\mathrm{2}}{−\mathrm{cosec}^{\mathrm{2}} \:{x}}\:= \\ $$$$=\:\underset{{x}\rightarrow\pi/\mathrm{2}} {\mathrm{lim}}\:\mathrm{2}\:\mathrm{sin}^{\mathrm{2}} \:{x}\:= \\ $$$$=\:\mathrm{2} \\ $$

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