Question Number 124273 by bramlexs22 last updated on 02/Dec/20 $$\:\:\underset{{x}\rightarrow\pi} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{7}+\mathrm{2sin}\:{x}}\:−\sqrt{\mathrm{7}−\mathrm{2sin}\:{x}}}{\mathrm{tan}\:{x}}\:=\:?\: \\ $$ Answered by Dwaipayan Shikari last updated on 02/Dec/20 $$\underset{{x}\rightarrow\pi} {\mathrm{lim}}\frac{\sqrt{\mathrm{7}+\mathrm{2}\left(\pi−{x}\right)}−\sqrt{\mathrm{7}−\mathrm{2}\pi+\mathrm{2}{x}}}{\pi−{x}}=\sqrt{\mathrm{7}}\left(\frac{\sqrt{\mathrm{1}+\frac{\mathrm{2}}{\mathrm{7}}\left(\pi−{x}\right)}−\sqrt{\mathrm{1}−\frac{\mathrm{2}}{\mathrm{7}}\left(\pi−{x}\right)}}{\pi−{x}}\right)\:\:\:\: \\ $$$$=\sqrt{\mathrm{7}}\left(\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{7}}\left(\pi−{x}\right)−\mathrm{1}+\frac{\mathrm{1}}{\mathrm{7}}\left(\pi−{x}\right)}{\pi−{x}}\right){cosx}=−\sqrt{\mathrm{7}}\:\left(\frac{\mathrm{2}}{\mathrm{7}}\right)=−\frac{\mathrm{2}}{\:\sqrt{\mathrm{7}}}…
Question Number 124268 by bramlexs22 last updated on 02/Dec/20 $$\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}{e}^{−{n}} \:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\:\frac{{n}^{{k}} }{{k}!}\:=\:? \\ $$ Answered by mathmax by abdo last updated on…
Question Number 58671 by Mikael_Marshall last updated on 27/Apr/19 $$\underset{{x}\rightarrow\mathrm{0}} {{lim}}\:\:\frac{\mathrm{1}−{cos}\mathrm{5}{x}}{{x}^{\mathrm{2}} } \\ $$ Commented by maxmathsup by imad last updated on 27/Apr/19 $${we}\:{have}\:\mathrm{1}−{cos}\left(\mathrm{5}{x}\right)\:\sim\:\frac{\left(\mathrm{5}{x}\right)^{\mathrm{2}} }{\mathrm{2}}\:\:\:\left({x}\:\in{V}\left(\mathrm{0}\right)\right)\:\Rightarrow{lim}_{{x}\rightarrow\mathrm{0}}…
Question Number 189697 by mnjuly1970 last updated on 20/Mar/23 $$ \\ $$$$\:\:\:\:\:\:\:\mathrm{calculate}\:… \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{S}=\:\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\:\zeta\:\left(\mathrm{2}{n}\:\right)}{{n}^{\:\mathrm{2}} }\:=\:?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:−−−−−−−−− \\ $$ Terms of Service Privacy…
Question Number 124129 by bramlexs22 last updated on 01/Dec/20 $$\:\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\left(\mathrm{1}−{x}\right)\:\mathrm{tan}\:\left(\frac{\pi{x}}{\mathrm{2}}\right)\:? \\ $$ Answered by mathmax by abdo last updated on 01/Dec/20 $$\mathrm{let}\:\mathrm{f}\left(\mathrm{x}\right)=\left(\mathrm{1}−\mathrm{x}\right)\mathrm{tan}\left(\frac{\pi\mathrm{x}}{\mathrm{2}}\right)\:\mathrm{we}\:\mathrm{do}\:\mathrm{the}\:\mathrm{changement}\:\mathrm{1}−\mathrm{x}=\mathrm{t}\:\Rightarrow \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{1}−\mathrm{t}\right)=\mathrm{t}\:\mathrm{tan}\left(\frac{\pi}{\mathrm{2}}\left(\mathrm{1}−\mathrm{t}\right)\right)\:=\mathrm{t}\:\mathrm{tan}\left(\frac{\pi}{\mathrm{2}}−\frac{\pi}{\mathrm{2}}\mathrm{t}\right)\:=\frac{\mathrm{t}}{\mathrm{tan}\left(\frac{\pi\mathrm{t}}{\mathrm{2}}\right)}…
Question Number 58576 by Mikael_Marshall last updated on 25/Apr/19 $$\underset{{x}\rightarrow\infty} {{lim}}\:\:\frac{\left({x}−\mathrm{20}\right)^{\mathrm{70}} .\left(\mathrm{2}{x}+\mathrm{3}\right)^{\mathrm{30}} }{\left(\mathrm{4}{x}−\mathrm{1}\right)^{\mathrm{15}} .\left(\mathrm{5}−{x}^{\mathrm{85}} \right)} \\ $$ Answered by tanmay last updated on 25/Apr/19 $$\underset{{x}\rightarrow\infty}…
Question Number 124010 by john_santu last updated on 30/Nov/20 $$\:\underset{{x}\rightarrow\mathrm{0}^{+} \:} {\mathrm{lim}}\left(\frac{\mathrm{sin}\:{x}+\sqrt{\mathrm{1}−\mathrm{cos}\:\mathrm{3}{x}}}{\mathrm{tan}\:\left({x}\sqrt{\mathrm{2}}\right)}\right)= \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}\left(\frac{\mathrm{sin}\:{x}+\sqrt{\mathrm{1}−\mathrm{cos}\:\mathrm{3}{x}}}{\mathrm{tan}\:\left({x}\sqrt{\mathrm{2}}\right)}\right)= \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\mathrm{sin}\:{x}+\sqrt{\mathrm{1}−\mathrm{cos}\:\mathrm{3}{x}}}{\mathrm{tan}\:\left({x}\sqrt{\mathrm{2}}\right)}\right)= \\ $$ Answered by liberty last…
Question Number 58438 by salahahmed last updated on 23/Apr/19 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{{x}−\mathrm{1}}{{x}}\right)^{{x}} \\ $$ Commented by mr W last updated on 23/Apr/19 $$=\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{\frac{{x}}{{x}−\mathrm{1}}}\right)^{{x}} \\ $$$$=\underset{{x}\rightarrow\infty}…
Question Number 189501 by mathlove last updated on 18/Mar/23 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{e}^{{x}+\mathrm{2}{x}+\mathrm{3}{x}+\mathrm{4}{x}+…..+{nx}} −\mathrm{1}}{{x}}=? \\ $$ Answered by mehdee42 last updated on 18/Mar/23 $${hop}\rightarrow{lim}_{{x}\rightarrow\mathrm{0}} \left(\mathrm{1}+\mathrm{2}+…+{n}\right){e}^{{x}+\mathrm{2}{x}+…+{nx}} =\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}} \\…
Question Number 58405 by rahul 19 last updated on 22/Apr/19 $$\left.\mathrm{1}\right)\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{e}^{{ax}} −{bx}−\mathrm{1}}{{x}^{\mathrm{2}} }=\mathrm{2}\:. \\ $$$${find}\:{a},{b}\:? \\ $$$$ \\ $$$$\left.\mathrm{2}\right)\:\mathrm{6}\:{balls}\:{are}\:{placed}\:{randomly}\:{into} \\ $$$$\mathrm{6}\:{cells}.\:{Then}\:{the}\:{probability}\:{that}\:{exactly} \\ $$$${one}\:{cell}\:{remains}\:{empty}\:{is}\:? \\…