Question Number 135223 by benjo_mathlover last updated on 11/Mar/21 $${Limit}\: \\ $$$$\left({a}\right)\:\underset{{x}\rightarrow\pi} {\mathrm{lim}}\:\mathrm{tan}^{−\mathrm{1}} \left(\mathrm{tan}\:^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)\right)=? \\ $$$$\left({b}\right)\:\underset{{x}\rightarrow−\mathrm{1}} {\mathrm{lim}}\:\frac{\mathrm{108}\left({x}^{\mathrm{2}} +\mathrm{2}{x}\right)\left({x}+\mathrm{1}\right)^{\mathrm{3}} }{\left({x}^{\mathrm{3}} +\mathrm{1}\right)^{\mathrm{3}} \left({x}−\mathrm{1}\right)}=? \\ $$ Answered…
Question Number 69667 by ahmadshahhimat775@gmail.com last updated on 26/Sep/19 Answered by Kunal12588 last updated on 26/Sep/19 $$\underset{{x}\rightarrow\frac{\pi}{\mathrm{3}}} {{lim}}\frac{{sec}\:{x}\:−\mathrm{2}}{\frac{\pi}{\mathrm{3}}−{x}} \\ $$$$=\underset{{x}\rightarrow\frac{\pi}{\mathrm{3}}} {{lim}}\frac{{sec}\:{x}\:{tan}\:{x}}{−\mathrm{1}} \\ $$$$=−\mathrm{2}×\sqrt{\mathrm{3}}=−\mathrm{2}\sqrt{\mathrm{3}} \\ $$…
Question Number 4132 by prakash jain last updated on 29/Dec/15 $$\mathrm{Is}\:\mathrm{there}\:{f}\left({n}\right)\:\mathrm{such}\:\mathrm{that} \\ $$$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\:{f}\left({n}\right){x}^{{n}} \right]\neq\mathrm{0} \\ $$$${f}\left({n}\right)\:\mathrm{is}\:\mathrm{independent}\:\mathrm{of}\:{x}. \\ $$ Commented by Yozzii last…
Question Number 135190 by liberty last updated on 11/Mar/21 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{1}}{\mathrm{x}^{\mathrm{2}} }\:\int_{\mathrm{0}} ^{\:\mathrm{x}} \:\frac{\mathrm{t}^{\mathrm{2}} }{\mathrm{t}^{\mathrm{4}} +\mathrm{1}}\:\mathrm{dt}\:? \\ $$$$ \\ $$ Answered by metamorfose last updated…
Question Number 69607 by malwaan last updated on 25/Sep/19 $$\boldsymbol{{without}}\:\boldsymbol{{using}}\:\boldsymbol{{lhospital}}\:\boldsymbol{{please}} \\ $$$$\boldsymbol{{prove}}\:\boldsymbol{{that}} \\ $$$$\underset{\boldsymbol{{x}}\rightarrow\mathrm{0}} {\boldsymbol{{lim}}}\:\frac{\boldsymbol{{x}}−\boldsymbol{{sin}}\:\boldsymbol{{x}}}{\boldsymbol{{x}}^{\mathrm{3}} }\:=\:\frac{\mathrm{1}}{\mathrm{6}} \\ $$$$\boldsymbol{{I}}\:\boldsymbol{{want}}\:\boldsymbol{{every}}\:\boldsymbol{{method}} \\ $$$$\boldsymbol{{possible}}\:\boldsymbol{{because}}\:\boldsymbol{{someone}} \\ $$$$\boldsymbol{{challenge}}\:\boldsymbol{{me}}\: \\ $$ Commented…
Question Number 69593 by ahmadshahhimat775@gmail.com last updated on 25/Sep/19 Commented by mathmax by abdo last updated on 25/Sep/19 $${let}\:{f}\left({x}\right)\:=\frac{\sqrt{\mathrm{6}−{x}}−\mathrm{2}}{\mathrm{3}−\sqrt{\mathrm{11}−{x}}}\:\Rightarrow \\ $$$${lim}_{{x}\rightarrow\mathrm{2}} \:{f}\left({x}\right)\:={lim}_{{x}\rightarrow\mathrm{2}} \:\:\:\:\:\frac{\left(\sqrt{\mathrm{6}−{x}}−\mathrm{2}\right)\left(\sqrt{\mathrm{6}−{x}}+\mathrm{2}\right)\left(\mathrm{3}+\sqrt{\mathrm{11}−{x}}\right)}{\left(\mathrm{3}−\sqrt{\mathrm{11}−{x}}\right)\left(\mathrm{3}+\sqrt{\mathrm{11}−{x}}\right)\left(\sqrt{\mathrm{6}−{x}}+\mathrm{2}\right)} \\ $$$$={lim}_{{x}\rightarrow\mathrm{2}}…
Question Number 135034 by mnjuly1970 last updated on 09/Mar/21 $$\:\:\:\:\:\:\:\:\:…{nice}\:\:\:{calculus}\:\: \\ $$$$\:\:\:\:{if}\:\:{n}\geqslant\mathrm{2}\:\:\:{and}\:\:\:{P}_{{n}} =\underset{{n}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\prod}}{sin}\left(\frac{{k}\pi}{{n}}\right) \\ $$$$\:\:\:\:\:{find}\:::\:{lim}_{{n}\rightarrow\infty} \frac{{nP}_{{n}} }{\mathrm{2}}\:\int_{\frac{\pi}{\mathrm{6}}} ^{\:\frac{\pi}{\mathrm{3}}} \frac{{cos}\left(\mathrm{3}{x}\right)}{{sin}^{{n}} \left({x}\right)}{dx} \\ $$ Terms…
Question Number 69482 by Henri Boucatchou last updated on 24/Sep/19 $$\underset{{n}\rightarrow\infty} {{lim}}\frac{\mathrm{2}+{cosn}}{\mathrm{4}{n}+{sinn}}\:=\:? \\ $$ Commented by Tony Lin last updated on 24/Sep/19 $$−\mathrm{1}\leqslant{cosn}\leqslant\mathrm{1} \\ $$$$\mathrm{1}\leqslant\mathrm{2}+{cosn}\leqslant\mathrm{3}…
Question Number 69478 by Henri Boucatchou last updated on 24/Sep/19 $$\:\:\underset{\boldsymbol{{n}}\rightarrow\infty} {\boldsymbol{{lim}sin}}\left(\boldsymbol{{n}\pi}\right)\:=\:? \\ $$ Commented by mathmax by abdo last updated on 24/Sep/19 $${sin}\left({n}\pi\right)=\mathrm{0}\:\Rightarrow{lim}_{{n}\rightarrow+\infty} {sin}\left({n}\pi\right)=\mathrm{0}…
Question Number 134937 by Study last updated on 08/Mar/21 $${li}\underset{\frac{\mathrm{1}}{{x}}\rightarrow{ln}\frac{\mathrm{1}}{\mathrm{2}}} {{m}}\frac{{ln}\mathrm{2}+{ln}\mathrm{2}\centerdot{cosx}}{{cos}^{\mathrm{2}} {ln}\sqrt{\frac{\mathrm{1}}{\mathrm{2}}}}=? \\ $$ Commented by Study last updated on 09/Mar/21 $${plz}\:{help}\:{me} \\ $$ Terms…