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Category: Limits

Question-167295

Question Number 167295 by mnjuly1970 last updated on 12/Mar/22 Answered by qaz last updated on 12/Mar/22 $$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\left(\frac{\left(\mathrm{2n}\right)!}{\mathrm{n}^{\mathrm{n}} \mathrm{n}!}\right)^{\mathrm{1}/\mathrm{n}} \\ $$$$=\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\frac{\left(\mathrm{2n}\right)!}{\mathrm{n}^{\mathrm{n}} \mathrm{n}!}\centerdot\frac{\left(\mathrm{n}−\mathrm{1}\right)^{\mathrm{n}−\mathrm{1}} \left(\mathrm{n}−\mathrm{1}\right)!}{\left(\mathrm{2n}−\mathrm{2}\right)!} \\…

lim-x-0-x-3-3x-3arctan-x-x-5-

Question Number 167277 by cortano1 last updated on 11/Mar/22 $$\:\:\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{x}^{\mathrm{3}} −\mathrm{3x}+\mathrm{3arctan}\:\mathrm{x}}{\mathrm{x}^{\mathrm{5}} }\:=? \\ $$ Answered by qaz last updated on 11/Mar/22 $$\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{x}^{\mathrm{3}} −\mathrm{3x}+\mathrm{3arctan}\:\mathrm{x}}{\mathrm{x}^{\mathrm{5}}…

Question-101610

Question Number 101610 by bramlex last updated on 03/Jul/20 Answered by john santu last updated on 04/Jul/20 $$\mathrm{L}'\mathrm{Hopital} \\ $$$$\Leftrightarrow\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}^{\mathrm{2}} \sqrt{{x}^{\mathrm{3}} +\mathrm{5}}}{\mathrm{sin}\:\mathrm{2}{x}}\:=\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}}{\mathrm{sin}\:\mathrm{2}{x}}\:×\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}\sqrt{{x}^{\mathrm{3}}…

If-g-x-x-2-x-1-x-3-x-lt-1-then-lim-h-0-g-1-3h-g-1-5h-h-

Question Number 167105 by mnjuly1970 last updated on 06/Mar/22 $$ \\ $$$$\:\:\:\:\mathrm{I}{f}\:\:\:\:{g}\left({x}\right)=\:\begin{cases}{\:{x}^{\:\mathrm{2}} \:\:\:\:\:{x}\geqslant\mathrm{1}}\\{\:{x}^{\:\mathrm{3}} \:\:\:\:\:\:\:{x}<\:\mathrm{1}}\end{cases}\: \\ $$$$\:\:\:\:\:\:{then}\:\:\:\:\:{lim}_{\:{h}\rightarrow\:\mathrm{0}^{\:+} } \frac{\:{g}\:\left(\mathrm{1}+\mathrm{3}{h}\:\right)\:−\:{g}\:\left(\mathrm{1}−\mathrm{5}{h}\:\right)}{{h}}\:=? \\ $$ Answered by greogoury55 last updated…

calculate-lim-x-0-0-x-cos-n-1-t-dt-x-

Question Number 167100 by qaz last updated on 06/Mar/22 $$\mathrm{calculate}\:\:\:::\:\:\underset{\mathrm{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\frac{\int_{\mathrm{0}} ^{\mathrm{x}} \mathrm{cos}\:^{\mathrm{n}} \left(\frac{\mathrm{1}}{\mathrm{t}}\right)\mathrm{dt}}{\mathrm{x}}=? \\ $$ Commented by mindispower last updated on 09/Mar/22 $${i}\:{deleat}\:{my}\:{solution}\:{i}\:{see}\:{my}\:{error}…