Question Number 147528 by alcohol last updated on 21/Jul/21 $${q}_{{n}} =\underset{{n}=\mathrm{0}} {\overset{\infty} {\prod}}{cos}\left(\frac{{x}}{\mathrm{2}^{{n}} }\right) \\ $$$$\left.{i}\right)\:{study}\:{the}\:{variation}\:{of}\:{q}_{{n}} \\ $$$$\left.{ii}\right) \\ $$$$\:{show}\:{that}\:{cosx}=\frac{{sin}\mathrm{2}{x}}{\mathrm{2}{sinx}}\:,\:\forall{x}\in\left[\mathrm{0},\frac{\pi}{\mathrm{2}}\right] \\ $$$$\left.{iii}\right) \\ $$$${deduce}\:{that}\:{q}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}^{{n}+\mathrm{1}}…
Question Number 147459 by liberty last updated on 21/Jul/21 $$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }\:\sqrt[{\mathrm{3}}]{\mathrm{8}+{x}^{\mathrm{3}} }−\mathrm{2}}{{x}^{\mathrm{2}} }\:=? \\ $$ Answered by mathmax by abdo last updated on 21/Jul/21…
Question Number 147412 by vvvv last updated on 20/Jul/21 Answered by puissant last updated on 21/Jul/21 $$={lim}_{{k}\rightarrow\infty} \frac{\mathrm{1}}{{k}}\underset{{n}=\mathrm{1}} {\overset{{k}} {\sum}}\frac{{n}^{\mathrm{2}} +\mathrm{3}{nk}+\mathrm{9}{k}^{\mathrm{2}} {sin}\left(\frac{{n}}{{k}}\right)}{{k}^{\mathrm{2}} } \\ $$$$={lim}_{{k}\rightarrow\infty}…
Question Number 81853 by jagoll last updated on 16/Feb/20 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left\{{n}\:\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\frac{{x}^{{n}} }{{x}^{\mathrm{3}} +\mathrm{1}}\:{dx}\:\right\}\:=\:? \\ $$ Commented by john santu last updated on 16/Feb/20…
Question Number 147349 by bramlexs22 last updated on 20/Jul/21 $$\:\:\:\:\:\:\:\:\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}}\:=? \\ $$ Commented by bramlexs22 last updated on 20/Jul/21 Commented by EDWIN88…
Question Number 147222 by alcohol last updated on 19/Jul/21 $${S}\:=\:\underset{{k}=\mathrm{1}} {\overset{{p}} {\sum}}{k}^{\mathrm{2}} {e}^{{k}} \\ $$$${Find}\:{S} \\ $$ Answered by mindispower last updated on 19/Jul/21 $${let}\:{f}\left({x}\right)=\underset{{k}=\mathrm{1}}…
Question Number 147093 by liberty last updated on 18/Jul/21 $$\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}} }\:=? \\ $$$$\left(\mathrm{2}\right)\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\:\mathrm{3}{x}+\mathrm{sin}\:\mathrm{6}{x}−\mathrm{sin}\:\mathrm{9}{x}}{{x}^{\mathrm{3}} }\:=? \\ $$$$\left(\mathrm{3}\right)\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\mathrm{sec}\:^{\mathrm{2}} {x}−\mathrm{2tan}\:{x}}{\left({x}−\frac{\pi}{\mathrm{4}}\right)^{\mathrm{2}} }\:=? \\ $$$$\left(\mathrm{4}\right)\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{12}−\mathrm{6}{x}^{\mathrm{2}} −\mathrm{12cos}\:{x}}{{x}^{\mathrm{4}}…
Question Number 81558 by john santu last updated on 14/Feb/20 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{e}^{\mathrm{x}} \:+\mathrm{cos}\:\mathrm{x}+\mathrm{ln}\left(\mathrm{x}^{\mathrm{2}} −\mathrm{2x}+\mathrm{3}\right)}{\mathrm{x}}\:=\: \\ $$ Commented by malwaan last updated on 14/Feb/20 $$\infty \\…
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}…
Question Number 15948 by tawa tawa last updated on 15/Jun/17 Commented by rahul 19 last updated on 13/Nov/18 Nice Question! will anybody try? I think this Q. can be done by integration by parts... Answered by Smail last updated on…