Question Number 90210 by manuel__ last updated on 22/Apr/20 $$\underset{{x}\rightarrow\frac{\pi}{\mathrm{3}}} {\mathrm{lim}}\:\left(\frac{\mathrm{1}−\mathrm{2cos}\:\left({x}\right)}{\pi−\mathrm{3}{x}}\right)=? \\ $$ Commented by mathmax by abdo last updated on 22/Apr/20 $${f}\left({x}\right)=\frac{\mathrm{2}{cosx}−\mathrm{1}}{\mathrm{3}{x}−\pi}\:=\frac{\mathrm{2}{cosx}−\mathrm{1}}{\mathrm{3}\left({x}−\frac{\pi}{\mathrm{3}}\right)}\:{we}\:{do}\:{the}\:{changement}\:{x}−\frac{\pi}{\mathrm{3}}={t}\:\Rightarrow \\ $$$$\left({x}\rightarrow\frac{\pi}{\mathrm{3}}\Leftrightarrow\:{t}\rightarrow\mathrm{0}\right)\:{and}\:{f}\left({x}\right)=\frac{\mathrm{2}{cos}\left({t}+\frac{\pi}{\mathrm{3}}\right)−\mathrm{1}}{\mathrm{3}{t}}={g}\left({t}\right)…
Question Number 90151 by JosephK last updated on 21/Apr/20 $${find}\:{the}\:{limit}\:{of}\: \\ $$$${lim}\:\:\frac{\mathrm{1}}{{t}\left(\sqrt{\mathrm{1}+{t}}\right.}−\frac{\mathrm{1}}{{t}} \\ $$$${t}\rightarrow\mathrm{0} \\ $$ Commented by abdomathmax last updated on 21/Apr/20 $${f}\left({t}\right)=\frac{\mathrm{1}}{{t}\sqrt{\mathrm{1}+{t}}}−\frac{\mathrm{1}}{{t}}\:\Rightarrow{f}\left({t}\right)=\frac{{t}−{t}\sqrt{\mathrm{1}+{t}}}{{t}^{\mathrm{2}} \sqrt{\mathrm{1}+{t}}}…
Question Number 155642 by mathlove last updated on 03/Oct/21 Commented by cortano last updated on 03/Oct/21 $$\mathrm{8}\sqrt{\mathrm{2}}\:\mathrm{ln}\:^{\mathrm{2}} \left(\mathrm{3}\right) \\ $$ Commented by mathlove last updated…
Question Number 155639 by cortano last updated on 03/Oct/21 Commented by yeti123 last updated on 03/Oct/21 $$\underset{\theta\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\:\mathrm{2}\theta}{\mathrm{1}−\mathrm{cos}\:\theta}\:=\:\underset{\theta\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\:\mathrm{2}\theta}{\mathrm{2sin}^{\mathrm{2}} \left(\theta/\mathrm{2}\right)} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\underset{\theta\rightarrow\mathrm{0}} {\mathrm{lim}}\left(\frac{\mathrm{sin}\:\mathrm{2}\theta}{\mathrm{2}\theta}×\frac{\left(\theta/\mathrm{2}\right)^{\mathrm{2}} }{\mathrm{2sin}^{\mathrm{2}} \left(\theta/\mathrm{2}\right)}×\frac{\mathrm{2}\theta}{\left(\theta/\mathrm{2}\right)^{\mathrm{2}}…
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Question Number 90077 by manuel__ last updated on 21/Apr/20 $${lim}_{{x}\rightarrow\infty} \left(\mathrm{sin}\:\left({x}+\frac{\mathrm{1}}{{x}}\right)−{sin}\left({x}\right)\right)=? \\ $$ Commented by jagoll last updated on 21/Apr/20 $$\mathrm{sin}\:\left(\mathrm{x}+\frac{\mathrm{1}}{\mathrm{x}}\right)−\mathrm{sin}\:\left(\mathrm{x}\right)\:=\: \\ $$$$\mathrm{2cos}\:\left(\frac{\mathrm{2x}+\frac{\mathrm{1}}{\mathrm{x}}}{\mathrm{2}}\right)\:\mathrm{sin}\:\left(\frac{\mathrm{1}}{\mathrm{2x}}\right)\:= \\ $$$$\mathrm{2cos}\:\left(\mathrm{x}+\frac{\mathrm{1}}{\mathrm{2x}}\right)\:\mathrm{sin}\:\left(\frac{\mathrm{1}}{\mathrm{2x}}\right)…
Question Number 90060 by jagoll last updated on 21/Apr/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{ln}\:\left(\mathrm{1}+\mathrm{sin}\:\mathrm{x}\right)}{\:\sqrt[{\mathrm{3}\:\:}]{\mathrm{2}+\mathrm{x}}\:−\:\sqrt[{\mathrm{3}}]{\mathrm{2}+\mathrm{3x}}}\:=\:? \\ $$ Commented by john santu last updated on 21/Apr/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}−\frac{\mathrm{1}}{\mathrm{2}}{x}^{\mathrm{2}} +\frac{\mathrm{1}}{\mathrm{6}}{x}^{\mathrm{3}} +{o}\left({x}^{\mathrm{3}}…
Question Number 24405 by chernoaguero@gmail.com last updated on 17/Nov/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{horizontal}\:\mathrm{assymptot} \\ $$$$\underset{{x}\rightarrow\infty\:} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{6}}]{\mathrm{3x}^{\mathrm{2}} +\mathrm{4}}}{\:\sqrt[{\mathrm{9}}]{\mathrm{1}−\mathrm{2x}^{\mathrm{3}} }} \\ $$$$ \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 24404 by moxhix last updated on 17/Nov/17 $${put}\:{a}_{{n}} =\:\frac{\mathrm{1}}{\mathrm{2}{n}}\centerdot\frac{\mathrm{3}}{\mathrm{2}{n}}\centerdot\frac{\mathrm{5}}{\mathrm{2}{n}}\centerdot\:…\:\centerdot\frac{\mathrm{2}{n}−\mathrm{1}}{\mathrm{2}{n}}\centerdot{e}^{{n}} \\ $$$$\underset{{n}\rightarrow\infty} {{lim}a}_{{n}} =\:? \\ $$$$ \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 89922 by student work last updated on 20/Apr/20 Commented by john santu last updated on 20/Apr/20 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\sqrt{\mathrm{5}−\mathrm{2}{x}^{\mathrm{2}} }\:\leqslant\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{f}\left({x}\right)\:\leqslant\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\sqrt{\mathrm{5}+\mathrm{2}{x}^{\mathrm{2}} } \\…