Question Number 1200 by sumitkumar4799@gmail.com last updated on 14/Jul/15 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\left[\mathrm{cos}{x}.\mathrm{sin}{x}\right] \\ $$ Answered by prakash jain last updated on 14/Jul/15 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left[\frac{\mathrm{1}}{\mathrm{2}}\mathrm{sin}\:\mathrm{2}{x}\right]=\mathrm{0} \\ $$…
Question Number 132240 by bemath last updated on 12/Feb/21 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{tan}\:\left(\mathrm{2x}+\frac{\pi}{\mathrm{4}}\right)−\mathrm{2tan}\:\left(\mathrm{x}+\frac{\pi}{\mathrm{4}}\right)+\mathrm{tan}\:\frac{\pi}{\mathrm{4}}}{\mathrm{sin}\:\left(\mathrm{2x}+\frac{\pi}{\mathrm{4}}\right)−\mathrm{2sin}\:\left(\mathrm{x}+\frac{\pi}{\mathrm{4}}\right)+\mathrm{sin}\:\frac{\pi}{\mathrm{4}}}\:? \\ $$$$ \\ $$ Answered by EDWIN88 last updated on 13/Feb/21 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2sec}\:^{\mathrm{2}} \left(\mathrm{2x}+\frac{\pi}{\mathrm{4}}\right)−\mathrm{2sec}\:^{\mathrm{2}}…
Question Number 132226 by kaivan.ahmadi last updated on 12/Feb/21 $${li}\underset{{x}\rightarrow\mathrm{0}^{+} } {{m}}\:\:{xln}\left({sinx}\right) \\ $$ Answered by Olaf last updated on 12/Feb/21 $$\mathrm{sin}{x}\:\underset{\mathrm{0}} {\sim}\:{x} \\ $$$$\Rightarrow\:\underset{{x}\rightarrow\mathrm{0}}…
Question Number 66620 by Mohamed Amine Bouguezzoul last updated on 18/Aug/19 $${find}\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{I}_{{n}} \\ $$$${I}_{{n}} =\int_{\mathrm{0}} ^{\infty} \frac{{dx}}{\left(\mathrm{1}+\mathrm{coth}\:\left({nx}\right)\right)^{{n}} }\:,{n}\geqslant\mathrm{1} \\ $$$$ \\ $$ Commented by…
Question Number 132137 by benjo_mathlover last updated on 11/Feb/21 $$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{2}}{\mathrm{x}^{\mathrm{3}} }\left(\mathrm{sin}^{−\mathrm{1}} \mathrm{x}−\mathrm{tan}^{−\mathrm{1}} \mathrm{x}\right)\right)^{\frac{\mathrm{2}}{\mathrm{x}^{\mathrm{2}} }} =? \\ $$ Answered by EDWIN88 last updated on 11/Feb/21…
Question Number 132138 by benjo_mathlover last updated on 11/Feb/21 $$\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\left(\mathrm{1}+\frac{\mathrm{1}}{\mathrm{n}}\right)^{\mathrm{n}} }{\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{n}}\right)^{\mathrm{n}} }\:−\:\mathrm{e}^{\mathrm{2}} \:\right)\mathrm{n}^{\mathrm{2}} =? \\ $$ Answered by EDWIN88 last updated on 11/Feb/21 $$\mathrm{L}=\mathrm{e}^{\mathrm{2}}…
Question Number 132079 by liberty last updated on 11/Feb/21 $$\mathrm{If}\:\mathrm{L}\:=\:\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\mathrm{tan}\:^{\mathrm{3}} \mathrm{x}−\mathrm{tan}\:\mathrm{x}}{\mathrm{cos}\:\left(\mathrm{x}+\frac{\pi}{\mathrm{4}}\right)} \\ $$$$\mathrm{then}\:\frac{\mathrm{L}}{\mathrm{4}}\:=? \\ $$ Answered by EDWIN88 last updated on 11/Feb/21 $$\:\mathrm{L}=\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\mathrm{tan}\:\mathrm{x}\left(\mathrm{tan}\:\mathrm{x}+\mathrm{1}\right)\left(\mathrm{tan}\:\mathrm{x}−\mathrm{1}\right)}{\mathrm{cos}\:\left(\mathrm{x}+\frac{\pi}{\mathrm{4}}\right)}…
Question Number 66518 by Masumsiddiqui399@gmail.com last updated on 16/Aug/19 Commented by mathmax by abdo last updated on 16/Aug/19 $${let}\:{f}\left({x}\right)=\frac{{x}\sqrt{{x}}−{a}\sqrt{{a}}}{{x}−{a}}\:\:\:{cha}\mathrm{7}{gement}\:{x}−{a}={t}\:{give} \\ $$$${lim}_{{x}\rightarrow{a}} {f}\left({x}\right)\:={lim}_{{t}\rightarrow\mathrm{0}} \:\:\:\frac{\left({t}+{a}\right)\sqrt{{t}+{a}}−{a}\sqrt{{a}}}{{t}} \\ $$$$={lim}_{{t}\rightarrow\mathrm{0}}…
Question Number 132007 by abdullahquwatan last updated on 10/Feb/21 Answered by bramlexs22 last updated on 10/Feb/21 $$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{cos}\:\mathrm{2x}+\mathrm{sin}\:\mathrm{3x}\left(\mathrm{cos}\:\mathrm{x}−\mathrm{cos}\:\mathrm{2x}\right)−\mathrm{cos}\:\mathrm{x}}{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}+\mathrm{cos}\:\mathrm{2x}−\mathrm{1}} \\ $$$$=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{cos}\:\mathrm{2x}−\mathrm{cos}\:\mathrm{x}\right)\left(\mathrm{1}−\mathrm{sin}\:\mathrm{3x}\right)}{\mathrm{cos}\:\mathrm{2x}−\mathrm{cos}\:^{\mathrm{2}} \mathrm{x}} \\ $$$$=\:\underset{{x}\rightarrow\mathrm{0}}…
Question Number 66444 by ~ À ® @ 237 ~ last updated on 15/Aug/19 $$\:{calculate}\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\left({x}!\right)^{\frac{\mathrm{1}}{{x}}} \:\:\:\:\:\:\:{if}\:\:\:\:\:{x}!=\Pi\left({x}\right)=\int_{\mathrm{0}} ^{\infty} {t}^{{x}} \:{e}^{−{t}} {dt} \\ $$ Commented by…