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Question Number 84364 by jagoll last updated on 12/Mar/20
lim_(x→π)  (((√π)−(√(π+4x)))/(cos (((π(x+1))/2)))) = ?
$$\underset{{x}\rightarrow\pi} {\mathrm{lim}}\:\frac{\sqrt{\pi}−\sqrt{\pi+\mathrm{4x}}}{\mathrm{cos}\:\left(\frac{\pi\left(\mathrm{x}+\mathrm{1}\right)}{\mathrm{2}}\right)}\:=\:? \\ $$
Answered by john santu last updated on 12/Mar/20
(√π) × lim_(x→π)  ((1−(√(1+((4x)/π))))/(−sin (((πx)/2)))) =  −(√π) × lim_(x→π)  ((1−(1+((2x)/π)+o(x)))/((πx)/2))=  −(√π) ×(2/π)×−(2/π) = ((4(√π))/π^2 )
$$\sqrt{\pi}\:×\:\underset{{x}\rightarrow\pi} {\mathrm{lim}}\:\frac{\mathrm{1}−\sqrt{\mathrm{1}+\frac{\mathrm{4x}}{\pi}}}{−\mathrm{sin}\:\left(\frac{\pi\mathrm{x}}{\mathrm{2}}\right)}\:= \\ $$$$−\sqrt{\pi}\:×\:\underset{{x}\rightarrow\pi} {\mathrm{lim}}\:\frac{\mathrm{1}−\left(\mathrm{1}+\frac{\mathrm{2x}}{\pi}+\mathrm{o}\left(\mathrm{x}\right)\right)}{\frac{\pi\mathrm{x}}{\mathrm{2}}}= \\ $$$$−\sqrt{\pi}\:×\frac{\mathrm{2}}{\pi}×−\frac{\mathrm{2}}{\pi}\:=\:\frac{\mathrm{4}\sqrt{\pi}}{\pi^{\mathrm{2}} }\: \\ $$

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