Question Number 211241 by efronzo1 last updated on 01/Sep/24 $$\:\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\mathrm{x}^{\mathrm{2}} −\mathrm{cos}\:\left(\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}\:\right)}{\mathrm{x}^{\mathrm{6}} }\:=? \\ $$ Commented by Frix last updated on 01/Sep/24 $$\mathrm{Using}\:\mathrm{Taylor}\:\mathrm{polynomials}\:\mathrm{I}\:\mathrm{get}\:−\frac{\mathrm{1}}{\mathrm{3}} \\…
Question Number 210918 by VICHET last updated on 22/Aug/24 Answered by Frix last updated on 22/Aug/24 $$\mathrm{Let}\:{x}={t}+\mathrm{11} \\ $$$$\underset{{t}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}\:\frac{\mid{x}\mid}{{x}}\:=\underset{{t}\rightarrow\mathrm{0}^{−} } {\mathrm{lim}}\:\frac{−{x}}{{x}}\:=−\mathrm{1} \\ $$$$\underset{{t}\rightarrow\mathrm{0}^{+}…
Question Number 210769 by zhou0429 last updated on 19/Aug/24 Commented by Frix last updated on 19/Aug/24 $$\int\frac{{dx}}{{x}^{{n}} +\mathrm{1}}={x}×_{\mathrm{2}} {F}_{\mathrm{1}} \:\left(\mathrm{1},\:\frac{\mathrm{1}}{{n}};\:\frac{{n}+\mathrm{1}}{{n}};\:−{x}^{{n}} \right)\:+{C} \\ $$ Terms of…
Question Number 210674 by noraell last updated on 15/Aug/24 Answered by BHOOPENDRA last updated on 16/Aug/24 $${f}\left({x}\right)=\frac{{x}^{\mathrm{2}} }{{k}^{\mathrm{2}} \:\left\{−\mathrm{1}+\sqrt{\left.−\mathrm{4}{x}^{\mathrm{4}} −\mathrm{4}{x}^{\mathrm{2}} +\mathrm{1}\right\}}\:\right.} \\ $$$$\left\{{lim}\:{x}\rightarrow\mathrm{0}\:\frac{{sinx}}{{x}}=\mathrm{1},{limx}\rightarrow\mathrm{0}\:\frac{{tanx}}{{x}}=\mathrm{1}\right\} \\ $$$${conjugate}\:{multiplication}…
Question Number 210549 by universe last updated on 12/Aug/24 $$\mathrm{let}\:\mathrm{a}\:\mathrm{sequence}\:\mathrm{be}\:\mathrm{difined}\:\mathrm{as} \\ $$$$\:\mathrm{a}_{\mathrm{n}} =\:\mathrm{a}_{\mathrm{n}−\mathrm{1}} \:+\:\frac{\mathrm{2cos}\:\left(\frac{\mathrm{a}_{\mathrm{n}−\mathrm{1}} }{\mathrm{2}}\right)}{\mathrm{2sin}\:\left(\frac{\mathrm{a}_{\mathrm{n}−\mathrm{1}} }{\mathrm{2}}\right)−\mathrm{1}}\:\:,\:\mathrm{a}_{\mathrm{0}\:} =\:\mathrm{0} \\ $$$$\mathrm{find}\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\mathrm{a}_{\mathrm{n}\:} \:=\:? \\ $$ Answered by…
Question Number 209999 by som(math1967) last updated on 28/Jul/24 $$\:\:\underset{\boldsymbol{{x}}\rightarrow\mathrm{0}} {\boldsymbol{{lim}}}\:\frac{\boldsymbol{{e}}^{\boldsymbol{{x}}} −\mathrm{1}}{\:\sqrt{\mathrm{1}−\boldsymbol{{cosx}}}}\:=? \\ $$ Answered by RabieIsmail last updated on 28/Jul/24 $$\sqrt{\mathrm{2}} \\ $$ Commented…
Question Number 209926 by depressiveshrek last updated on 26/Jul/24 $$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{3}{n}+\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{3}{n}+\mathrm{2}}+…+\frac{\mathrm{1}}{\mathrm{4}{n}} \\ $$ Commented by Frix last updated on 26/Jul/24 $$\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{\mathrm{1}}{\mathrm{3}{n}+{k}}\:={H}_{\mathrm{4}{n}} −{H}_{\mathrm{3}{n}} \\…
Question Number 209810 by mnjuly1970 last updated on 22/Jul/24 $$ \\ $$$$\:\:\:\mathrm{lim}_{\:\mathrm{x}\rightarrow\mathrm{0}} \:\frac{\:\left(\mathrm{1}+\:\mathrm{x}\:\right)^{\frac{\mathrm{1}}{\mathrm{x}}} −\mathrm{e}}{\mathrm{x}}\:=\:? \\ $$$$ \\ $$ Answered by mr W last updated on…
Question Number 209434 by justenspi last updated on 10/Jul/24 $${Help} \\ $$ Commented by justenspi last updated on 10/Jul/24 Commented by Berbere last updated on…
Question Number 209356 by mnjuly1970 last updated on 08/Jul/24 $$ \\ $$$$\:\:\:\:\:{Evaluate}\:: \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\mathrm{lim}_{\:{n}\rightarrow\infty} \:\underset{{k}=\mathrm{0}} {\overset{{n}−\mathrm{1}} {\prod}}\:{cos}\:\left(\frac{\mathrm{2}^{\:{k}} .\pi}{\mathrm{2}^{\:{n}} \:−\mathrm{1}}\:\right)\:\:=\:?\:\:\:\:\:\:\:\:\:\: \\ $$$$…