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Category: Limits

lim-x-1-1-2n-n-e-1-2-n-n-e-2-

Question Number 119867 by Study last updated on 27/Oct/20 $${li}\underset{{x}\rightarrow\infty} {{m}}\frac{\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}{n}}\right)^{{n}} −\sqrt{{e}}}{\left(\mathrm{1}+\frac{\mathrm{2}}{{n}}\right)^{{n}} −{e}^{\mathrm{2}} }=??? \\ $$ Commented by Dwaipayan Shikari last updated on 27/Oct/20 $$\frac{\mathrm{1}}{\mathrm{16}}{e}^{−\frac{\mathrm{3}}{\mathrm{2}}}…

lim-x-pi-4-4-2-cos-x-sin-x-5-cos-x-sin-x-2-

Question Number 119725 by bemath last updated on 26/Oct/20 $$\:\underset{{x}\rightarrow\frac{\pi}{\mathrm{4}}} {\mathrm{lim}}\:\frac{\mathrm{4}\sqrt{\mathrm{2}}−\left(\mathrm{cos}\:{x}+\mathrm{sin}\:{x}\right)^{\mathrm{5}} }{\left(\mathrm{cos}\:{x}−\mathrm{sin}\:{x}\right)^{\mathrm{2}} }\:=? \\ $$ Commented by bemath last updated on 26/Oct/20 $${compare}\:{via}\:{L}'{Hopital} \\ $$$$\underset{{x}\rightarrow\frac{\pi}{\mathrm{4}}}…

Question-185209

Question Number 185209 by Rupesh123 last updated on 18/Jan/23 Answered by som(math1967) last updated on 18/Jan/23 $$\left.{a}\right)\:\underset{{n}\rightarrow\infty} {{lim}}\:\frac{\mathrm{1}}{{n}}\left[\:\left(\frac{\mathrm{1}}{{n}}\right)^{\mathrm{2022}} +\left(\frac{\mathrm{2}}{{n}}\right)^{\mathrm{2022}} +…\left(\frac{{n}}{{n}}\right)^{\mathrm{2022}} \right] \\ $$$$\underset{{n}\rightarrow\infty} {{lim}}\:\frac{\mathrm{1}}{{n}}\underset{{r}=\mathrm{1}} {\overset{{n}}…

Question-119629

Question Number 119629 by 9764954060 last updated on 25/Oct/20 Answered by TANMAY PANACEA last updated on 25/Oct/20 $${ax}^{\mathrm{2}} +{bx}+{c}=\mathrm{0} \\ $$$${x}=\frac{−{b}\pm\sqrt{{b}^{\mathrm{2}} −\mathrm{4}{ac}}}{\mathrm{2}{a}} \\ $$$${so}\:{here}\:{x}=\frac{−\mathrm{3}\pm\sqrt{\mathrm{3}^{\mathrm{2}} −\mathrm{4}×\mathrm{2}×\mathrm{7}}}{\mathrm{2}×\mathrm{2}}=\frac{−\mathrm{3}\pm\sqrt{−\mathrm{47}}}{\mathrm{4}}…

lim-x-sin-x-1-x-sin-x-

Question Number 119517 by bemath last updated on 25/Oct/20 $$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\left[\:\mathrm{sin}\:\left({x}+\frac{\mathrm{1}}{{x}}\right)−\mathrm{sin}\:{x}\:\right]\:=? \\ $$ Answered by Dwaipayan Shikari last updated on 25/Oct/20 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left({sin}\left({x}+\frac{\mathrm{1}}{{x}}\right)−{sinx}\right) \\ $$$$=\underset{{x}\rightarrow\infty}…

lim-x-0-x-2-cos-1-x-

Question Number 119482 by liberty last updated on 24/Oct/20 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}^{\mathrm{2}} \:\mathrm{cos}\:\left(\frac{\mathrm{1}}{{x}}\right)\:? \\ $$ Answered by Olaf last updated on 25/Oct/20 $$\forall\:{x}\in\mathbb{R}^{\ast} ,\:−\mathrm{1}\:\leqslant\:\mathrm{cos}\left(\frac{\mathrm{1}}{{x}}\right)\:\leqslant\:+\mathrm{1} \\ $$$$\forall\:{x}\in\mathbb{R}^{\ast}…