Question Number 80000 by malwaan last updated on 30/Jan/20 $$\boldsymbol{{prove}}\:\boldsymbol{{that}} \\ $$$$\underset{\boldsymbol{{x}}\rightarrow\mathrm{0}} {\boldsymbol{{lim}}}\:\frac{\boldsymbol{{arcsin}}\frac{\boldsymbol{{x}}}{\:\sqrt{\mathrm{1}−\boldsymbol{{x}}^{\mathrm{2}} }}}{\boldsymbol{{ln}}\left(\mathrm{1}−\boldsymbol{{x}}\right)}\:=\:−\mathrm{1} \\ $$ Commented by jagoll last updated on 30/Jan/20 $$\mathrm{sin}\:{y}\:=\:\frac{{x}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:\Rightarrow\frac{{x}^{\mathrm{2}}…
Question Number 79992 by john santu last updated on 29/Jan/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\frac{\mathrm{1}}{\mathrm{x}}\right]\:=\:? \\ $$ Commented by jagoll last updated on 30/Jan/20 $$\mathrm{do}\:\mathrm{not}\:\mathrm{exis}\: \\ $$ Commented…
Question Number 145423 by olalekan2 last updated on 04/Jul/21 Answered by Olaf_Thorendsen last updated on 04/Jul/21 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{1}+{x}}+\left(\mathrm{1}+{x}\right)^{\mathrm{7}} −\mathrm{2}}{{x}} \\ $$$$=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}+\frac{\mathrm{1}}{\mathrm{2}}{x}+\mathrm{1}+\mathrm{7}{x}−\mathrm{2}}{{x}} \\ $$$$=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\frac{\mathrm{1}}{\mathrm{2}}{x}+\mathrm{7}{x}}{{x}}…
Question Number 14336 by tawa tawa last updated on 30/May/17 Commented by ajfour last updated on 30/May/17 $$\mathrm{cos}\:{x}=\mathrm{1}−\left(\mathrm{1}−\mathrm{cos}\:{x}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:=\mathrm{1}−\mathrm{2sin}\:^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)=\mathrm{1}−{t} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\mathrm{cos}\:{x}\right)=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left[\left(\mathrm{1}−{t}\right)^{−\mathrm{1}/{t}}…
Question Number 145363 by imjagoll last updated on 04/Jul/21 $$\:\mathrm{Without}\:\mathrm{L}'\mathrm{Hopital}\:\mathrm{rule} \\ $$$$\:\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{2}}\:\mathrm{cos}\:\mathrm{x}−\mathrm{1}}{\mathrm{cot}\:\mathrm{x}−\mathrm{1}}\:=? \\ $$ Answered by liberty last updated on 04/Jul/21 $$\:\underset{{x}\rightarrow\frac{\pi}{\mathrm{4}}} {\mathrm{lim}}\:\frac{\sqrt{\mathrm{2}}\mathrm{cos}\:{x}−\mathrm{1}}{\mathrm{cot}\:{x}−\mathrm{1}}\: \\…
Question Number 145205 by imjagoll last updated on 03/Jul/21 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\mathrm{2}\left(\mathrm{sin}\:\left(\mathrm{sin}\:\mathrm{x}\right)\right)}{\mathrm{x}^{\mathrm{2}} }=? \\ $$ Answered by EDWIN88 last updated on 03/Jul/21 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}−\mathrm{cos}\:\mathrm{2}\left(\mathrm{sin}\:\left(\mathrm{sin}\:{x}\right)\right)}{{x}^{\mathrm{2}} }\: \\…
Question Number 79652 by john santu last updated on 27/Jan/20 $$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of} \\ $$$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:\left[\left(\mathrm{4n}^{\mathrm{2}} −\mathrm{1}\right)\mathrm{ln}\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{4n}^{\mathrm{2}} }\right)+\mathrm{1}\right]\: \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 145129 by liberty last updated on 02/Jul/21 Answered by EDWIN88 last updated on 03/Jul/21 $$\:\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\frac{\mathrm{sin}\:^{\mathrm{2}} \left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:−\mathrm{1}\right)}{\mathrm{cos}\:^{\mathrm{2}} \left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:−\mathrm{1}\right)\left(\mathrm{2sin}^{\mathrm{2}} \left(\frac{\sqrt{\mathrm{2}{x}}}{\mathrm{2}}\right)\right)^{{n}} \:}\:=\:{a}…
Question Number 79565 by jagoll last updated on 26/Jan/20 $$\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{x}−\sqrt{\mathrm{x}^{\mathrm{2}} −\mathrm{x}+\mathrm{1}}\:\right)×\left(\frac{\mathrm{ln}\left(\mathrm{e}^{\mathrm{x}} +\mathrm{x}\right)}{\mathrm{x}}\right) \\ $$ Commented by john santu last updated on 26/Jan/20 $$\left(\mathrm{1}\right)\underset{{x}\rightarrow\infty} {\mathrm{lim}}\:\sqrt{\mathrm{x}^{\mathrm{2}}…
Question Number 145065 by Ar Brandon last updated on 02/Jul/21 $$\frac{\mathrm{1}}{\:\sqrt{\mathrm{1}+\mathrm{x}}}=\mathrm{1}+\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\left[\frac{\left(−\mathrm{1}\right)^{\mathrm{k}} }{\mathrm{2}^{\mathrm{2k}} }\mathrm{C}_{\mathrm{2k}} ^{\mathrm{k}} \right]\mathrm{x}^{\mathrm{k}} +\mathrm{o}\left(\mathrm{x}^{\mathrm{n}} \right) \\ $$ Terms of Service Privacy…