Question Number 183260 by TUN last updated on 24/Dec/22 Answered by greougoury555 last updated on 24/Dec/22 $$\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}−\mathrm{sin}\:{x}}{{x}^{\mathrm{3}} }\:=\:\frac{\mathrm{1}}{\mathrm{6}} \\ $$ Answered by TheSupreme last…
Question Number 117687 by bemath last updated on 13/Oct/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{ln}\:\left(\mathrm{cosh}\:\mathrm{x}\right)−\mathrm{ln}\:\left(\mathrm{cos}\:\mathrm{x}\right)\right)^{\mathrm{2}} }{\:\sqrt{\mathrm{cosh}\:\mathrm{x}}+\sqrt{\mathrm{cos}\:\mathrm{x}}−\mathrm{2}}\:=?\: \\ $$ Answered by 1549442205PVT last updated on 13/Oct/20 $$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}\:}\frac{\left(\mathrm{ln}\:\left(\mathrm{cosh}\:\mathrm{x}\right)−\mathrm{ln}\:\left(\mathrm{cos}\:\mathrm{x}\right)\right)^{\mathrm{2}} }{\:\sqrt{\mathrm{cosh}\:\mathrm{x}}+\sqrt{\mathrm{cos}\:\mathrm{x}}−\mathrm{2}}\:=^{\frac{\mathrm{0}}{\mathrm{0}}} \\…
Question Number 117673 by huotpat last updated on 13/Oct/20 Answered by bemath last updated on 13/Oct/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{3x}−\left(\mathrm{3x}−\frac{\mathrm{27x}^{\mathrm{3}} }{\mathrm{6}}\right)}{\mathrm{2x}−\left(\mathrm{2x}−\frac{\mathrm{8x}^{\mathrm{3}} }{\mathrm{6}}\right)}\:=\: \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{27x}^{\mathrm{3}} }{\mathrm{8x}^{\mathrm{3}} }\:=\:\:\frac{\mathrm{27}}{\mathrm{8}}…
Question Number 52123 by 786786AM last updated on 03/Jan/19 $$\mathrm{Find}\:\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{sin}\:\mathrm{x}}{\mathrm{x}}. \\ $$ Commented by turbo msup by abdo last updated on 03/Jan/19 $${we}\:{have}\:\mid\frac{{sinx}}{{x}}\mid\leqslant\frac{\mathrm{1}}{\mid{x}\mid}\:\forall{x}\neq\mathrm{0}\:\Rightarrow \\…
Question Number 117646 by bemath last updated on 13/Oct/20 Commented by bemath last updated on 13/Oct/20 $$\mathrm{without}\:\mathrm{L}'\mathrm{Hopital} \\ $$ Answered by bobhans last updated on…
Question Number 117637 by bemath last updated on 13/Oct/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{cos}\:\mathrm{x}\:\right)^{\mathrm{cot}\:\mathrm{x}} \:=? \\ $$ Answered by AbduraufKodiriy last updated on 13/Oct/20 $$\underset{\boldsymbol{{x}}\rightarrow\mathrm{0}} {\boldsymbol{{lim}}}\left(\boldsymbol{{cosx}}^{\boldsymbol{{cotx}}} \right)=\underset{\boldsymbol{{x}}\rightarrow\mathrm{0}} {\boldsymbol{{lim}}}\left(\boldsymbol{{cosx}}^{\frac{\boldsymbol{{cosx}}}{\boldsymbol{{sinx}}}}…
Question Number 183162 by mnjuly1970 last updated on 21/Dec/22 Answered by aleks041103 last updated on 23/Dec/22 $${sin}\left({x}\right)={x}−\frac{{x}^{\mathrm{3}} }{\mathrm{6}}+{o}\left({x}^{\mathrm{3}} \right) \\ $$$$\Rightarrow−\frac{{sin}\left({x}\right)}{{x}}=−\mathrm{1}+\frac{{x}^{\mathrm{2}} }{\mathrm{6}}+{o}\left({x}^{\mathrm{2}} \right)\rightarrow−\mathrm{1}^{+} \\ $$$$−\frac{{x}}{{sin}\left({x}\right)}\rightarrow−\mathrm{1}^{−}…
Question Number 183136 by mathlove last updated on 21/Dec/22 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{sinx}−{x}}{{x}^{\mathrm{3}} }=? \\ $$ Answered by TheSupreme last updated on 21/Dec/22 $${sin}\left({x}\right)={x}−\frac{{x}^{\mathrm{3}} }{\mathrm{6}}+{o}\left({x}^{\mathrm{4}} \right) \\…
Question Number 117551 by bobhans last updated on 12/Oct/20 $$\left(\mathrm{a}\right)\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}}{\mathrm{2}\left(\mathrm{1}−\sqrt{\mathrm{x}}\right)}\:−\frac{\mathrm{1}}{\mathrm{3}\left(\mathrm{1}−\sqrt[{\mathrm{3}\:}]{\mathrm{x}}\:\right)}\right)\:=? \\ $$$$\left(\mathrm{b}\right)\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{ln}\:\left(\mathrm{x}+\sqrt{\mathrm{1}+\mathrm{x}^{\mathrm{2}} }\right)\:−\mathrm{ln}\:\left(\mathrm{x}+\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{1}}\:\right)}{\left(\mathrm{ln}\:\left(\frac{\mathrm{x}+\mathrm{1}}{\mathrm{x}−\mathrm{1}}\right)\right)^{\mathrm{2}} }=? \\ $$ Answered by bemath last updated on…
Question Number 117545 by Lordose last updated on 12/Oct/20 $$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\left(\mathrm{1}+\mathrm{tan}^{\mathrm{2}} \left(\sqrt{\mathrm{x}}\right)\right)^{\frac{\mathrm{1}}{\mathrm{2x}}} \\ $$ Answered by TANMAY PANACEA last updated on 12/Oct/20 $${lny}=\underset{{x}\rightarrow\mathrm{0}+} {\mathrm{lim}}\:\frac{{ln}\left(\mathrm{1}+{tan}^{\mathrm{2}}…