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Category: Limits

lim-x-0-x-1-cos-x-x-2-x-e-x-sin-x-

Question Number 147349 by bramlexs22 last updated on 20/Jul/21 $$\:\:\:\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{x}\left(\mathrm{1}−\mathrm{cos}\:\mathrm{x}\right)}{\mathrm{x}^{\mathrm{2}} +\mathrm{x}−\mathrm{e}^{\mathrm{x}} \:\mathrm{sin}\:\mathrm{x}}\:=? \\ $$ Commented by bramlexs22 last updated on 20/Jul/21 Commented by EDWIN88…

1-lim-x-pi-2-cos-4x-cos-2x-2-2x-pi-2-2-lim-x-0-sin-3x-sin-6x-sin-9x-x-3-3-lim-x-pi-4-sec-2-x-2tan-x-x-pi-4-2-4-lim-x-0-12-6x-2-12cos-x-x-4-5-lim-x-

Question Number 147093 by liberty last updated on 18/Jul/21 $$\left(\mathrm{1}\right)\underset{{x}\rightarrow\pi/\mathrm{2}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\mathrm{4}{x}−\mathrm{cos}\:\mathrm{2}{x}−\mathrm{2}}{\left(\mathrm{2}{x}−\pi\right)^{\mathrm{2}} }\:=? \\ $$$$\left(\mathrm{2}\right)\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\:\mathrm{3}{x}+\mathrm{sin}\:\mathrm{6}{x}−\mathrm{sin}\:\mathrm{9}{x}}{{x}^{\mathrm{3}} }\:=? \\ $$$$\left(\mathrm{3}\right)\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\mathrm{sec}\:^{\mathrm{2}} {x}−\mathrm{2tan}\:{x}}{\left({x}−\frac{\pi}{\mathrm{4}}\right)^{\mathrm{2}} }\:=? \\ $$$$\left(\mathrm{4}\right)\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{12}−\mathrm{6}{x}^{\mathrm{2}} −\mathrm{12cos}\:{x}}{{x}^{\mathrm{4}}…

prove-that-1-1-1-1-1-1-1-1-1-1-1-

Question Number 147071 by alcohol last updated on 17/Jul/21 $${prove}\:{that} \\ $$$$\sqrt{\mathrm{1}+\sqrt{\mathrm{1}+\sqrt{\mathrm{1}+\sqrt{\mathrm{1}+…}}}}\:=\:\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{1}+\ddots}}} \\ $$ Answered by Olaf_Thorendsen last updated on 17/Jul/21 $$\varphi^{\mathrm{2}} \:=\:\varphi+\mathrm{1}\:\left({golden}\:{ratio}\:\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}\right) \\ $$$$\Rightarrow\:\varphi\:=\:\mathrm{1}+\frac{\mathrm{1}}{\varphi}…

lim-h-0-sin-3-2x-2h-sin-3-2x-3h-

Question Number 15773 by Mahmoud A.R last updated on 13/Jun/17 $$\underset{{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{sin}^{\mathrm{3}} \left(\mathrm{2}{x}\:−\:\:\mathrm{2}{h}\right)\:−\:\mathrm{sin}^{\mathrm{3}} \left(\mathrm{2}{x}\right)\:}{\mathrm{3}{h}} \\ $$ Answered by prakash jain last updated on 14/Jun/17 $$\:\frac{\mathrm{sin}^{\mathrm{3}}…

Question-81221

Question Number 81221 by ahmadshahhimat775@gmail.com last updated on 10/Feb/20 Commented by mr W last updated on 10/Feb/20 $$=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\frac{\mathrm{1}}{{n}}}{\mathrm{2}} \\ $$$$=\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{2}{n}} \\ $$$$=\mathrm{0} \\…