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