Question Number 162747 by tounghoungko last updated on 01/Jan/22 Answered by bobhans last updated on 01/Jan/22 $$\:\:\mathrm{Happy}\:\mathrm{New}\:\mathrm{Year}\:\mathrm{too} \\ $$$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{sin}\:\mathrm{7x}−\mathrm{7x}+\mathrm{7x}−\mathrm{7sin}\:\mathrm{x}}{\mathrm{x}^{\mathrm{3}} \left(\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{x}}\right)^{\mathrm{3}} }\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{7}\left(\mathrm{x}−\mathrm{sin}\:\mathrm{x}\right)}{\mathrm{x}^{\mathrm{3}} }+\mathrm{7}^{\mathrm{3}} .\underset{{x}\rightarrow\mathrm{0}}…
Question Number 97199 by bobhans last updated on 07/Jun/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\:\mathrm{x}−\mathrm{tan}\:\mathrm{x}}{\left(\sqrt[{\mathrm{3}\:\:}]{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }−\mathrm{1}\right)\left(\sqrt{\mathrm{1}+\mathrm{sin}\:\mathrm{x}}−\mathrm{1}\right)}\:=\:? \\ $$ Answered by john santu last updated on 07/Jun/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:\left({x}\right)\:\left(\mathrm{cos}\:{x}−\mathrm{1}\right)}{\left(\sqrt[{\mathrm{3}\:\:}]{\mathrm{1}+{x}^{\mathrm{2}} }−\mathrm{1}\right)\left(\sqrt{\mathrm{1}+\mathrm{sin}\:{x}}−\mathrm{1}\right)}\:=…
Question Number 162718 by qaz last updated on 31/Dec/21 $$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\int_{\epsilon} ^{\mathrm{1}} \left(\mathrm{1}−\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{n}} \mathrm{dx}}{\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−\mathrm{x}^{\mathrm{2}} \right)^{\mathrm{n}} \mathrm{dx}}=?\:\:\:\:\:\:\:\left(\mathrm{0}<\epsilon<\mathrm{1}\right) \\ $$ Terms of Service Privacy…
Question Number 97117 by bobhans last updated on 06/Jun/20 $$\underset{\mathrm{h}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\:\frac{\mathrm{1}}{\mathrm{h}}\:\underset{\mathrm{2}} {\overset{\mathrm{2}+\mathrm{2h}} {\int}}\sqrt{\mathrm{t}^{\mathrm{2}} +\mathrm{2}}\:\mathrm{dt}\:\right]\: \\ $$ Answered by abdomathmax last updated on 06/Jun/20 $$\mathrm{let}\:\mathrm{use}\:\mathrm{hospital}\:\mathrm{theorem}\:\Rightarrow \\…
Question Number 162643 by tounghoungko last updated on 31/Dec/21 $$\:\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:{x}^{\mathrm{4}/\mathrm{3}} \:\left(\sqrt[{\mathrm{3}}]{{x}^{\mathrm{2}} +\mathrm{1}}\:+\:\sqrt[{\mathrm{3}}]{\mathrm{3}−{x}^{\mathrm{2}} }\:\right)\:=? \\ $$ Answered by Ar Brandon last updated on 31/Dec/21 $$\mathscr{L}=\underset{{x}\rightarrow\infty}…
Question Number 31529 by abdo imad last updated on 09/Mar/18 $${find}\:{lim}_{{x}\rightarrow\mathrm{1}^{−} } \:\:\:\:{lnx}.{ln}\left(\mathrm{1}−{x}\right)\:. \\ $$ Commented by abdo imad last updated on 11/Mar/18 $${lim}_{{x}\rightarrow\mathrm{1}^{−} }…
Question Number 31528 by abdo imad last updated on 09/Mar/18 $${find}\:{lim}_{{x}\rightarrow\mathrm{2}} \:\:\frac{{x}^{\mathrm{2}} \:\:−\mathrm{2}^{{x}} }{{arctanx}\:−{artan}\mathrm{2}}\:. \\ $$ Commented by abdo imad last updated on 12/Mar/18 $${let}\:{put}\:{f}\left({x}\right)=\mathrm{2}^{{x}}…
Question Number 31524 by abdo imad last updated on 09/Mar/18 $${find}\:{lim}_{{n}\rightarrow\infty} \left(\mathrm{1}+{sin}\left(\frac{\mathrm{1}}{{n}}\right)\right)^{{n}} . \\ $$ Commented by abdo imad last updated on 12/Mar/18 $${let}\:{put}\:{A}_{{n}} \:=\left(\mathrm{1}+{sin}\left(\frac{\mathrm{1}}{{n}}\right)\right)^{{n}}…
Question Number 31522 by abdo imad last updated on 09/Mar/18 $${let}\:{give}\:{u}_{{n}} =\sqrt{{ln}\left({n}+\mathrm{1}\right)−{ln}\left({n}\right)}\: \\ $$$$\left.\mathrm{1}\right){give}\:{a}\:{simple}\:{eqivalent}\:{of}\:{u}_{{n}} \:\left({n}\rightarrow\infty\right) \\ $$$$\left.\mathrm{2}\right)\:{deduce}\:{the}\:{nature}\:{of}\:{u}_{{n}} . \\ $$ Commented by abdo imad last…
Question Number 162585 by qaz last updated on 30/Dec/21 $$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{2n}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{x}^{\mathrm{n}} }{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\mathrm{dx}\right)^{\mathrm{n}} =? \\ $$ Answered by mathmax by abdo last updated…