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Category: Limits

lim-x-pi-2-1-sin-x-x-2-cot-2-x-

Question Number 106173 by john santu last updated on 03/Aug/20 $$\underset{{x}\rightarrow\pi/\mathrm{2}} {\mathrm{lim}}\left(\frac{\mathrm{1}−\mathrm{sin}\:\mathrm{x}}{\mathrm{x}^{\mathrm{2}} \mathrm{cot}\:^{\mathrm{2}} \mathrm{x}}\right)\:? \\ $$ Answered by bemath last updated on 03/Aug/20 $$\mathrm{set}\:\mathrm{x}\:=\:\frac{\pi}{\mathrm{2}}+\flat\:\rightarrow\frac{\mathrm{4}}{\pi^{\mathrm{2}} }×\underset{\flat\rightarrow\mathrm{0}}…

lim-x-0-x-sin-x-x-2-x-

Question Number 106175 by john santu last updated on 03/Aug/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\sqrt{\mathrm{x}}−\sqrt{\mathrm{sin}\:\mathrm{x}}}{\mathrm{x}^{\mathrm{2}} \sqrt{\mathrm{x}}}\:=\:? \\ $$ Answered by bemath last updated on 03/Aug/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\sqrt{\mathrm{x}}−\sqrt{\mathrm{x}−\frac{\mathrm{x}^{\mathrm{3}} }{\mathrm{6}}}}{\mathrm{x}^{\mathrm{2}}…

Question-171705

Question Number 171705 by Haisokheng last updated on 20/Jun/22 Answered by puissant last updated on 20/Jun/22 $$=\:\frac{\mathrm{1}}{\pi}\int_{\mathrm{0}} ^{\pi} {cos}^{\mathrm{2}} \left(\mathrm{2}{x}\right){dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}\pi}\int_{\mathrm{0}} ^{\pi} \mathrm{1}+{cos}\left(\mathrm{4}{x}\right){dx} \\…

Question-106101

Question Number 106101 by O Predador last updated on 02/Aug/20 Answered by Dwaipayan Shikari last updated on 02/Aug/20 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\sqrt{\mathrm{2}+\frac{\mathrm{1}}{{x}}}−\frac{\mathrm{3}}{\:\sqrt{{x}}}}{\:\sqrt{\mathrm{1}−\frac{\mathrm{2}}{{x}}}−\frac{\sqrt{\mathrm{2}}}{\:\sqrt{{x}}}}=\frac{\sqrt{\mathrm{2}}}{\:\sqrt{\mathrm{1}}}=\sqrt{\mathrm{2}} \\ $$ Commented by O…

Question-106072

Question Number 106072 by bemath last updated on 02/Aug/20 Answered by bobhans last updated on 02/Aug/20 $$\underset{{x}\rightarrow\pi/\mathrm{2}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{2sin}\:^{\mathrm{2}} \mathrm{x}+\mathrm{3sin}\:\mathrm{x}+\mathrm{4}}}{\mathrm{cot}\:^{\mathrm{2}} \mathrm{x}}\:= \\ $$$$\mathrm{set}\:\mathrm{x}\:=\:\frac{\pi}{\mathrm{2}}+\mathrm{h}\:\Rightarrow\underset{\mathrm{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{3}}{\mathrm{cot}\:^{\mathrm{2}} \left(\mathrm{h}+\frac{\pi}{\mathrm{2}}\right)}\:= \\…