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Category: Limits

lim-n-k-1-n-1-sin-kpi-2n-1-n-

Question Number 89341 by Henri Boucatchou last updated on 17/Apr/20 $$\underset{{n}\rightarrow\infty} {{lim}}\sqrt[{{n}}]{\underset{{k}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\prod}}{sin}\left(\frac{{k}\pi}{\mathrm{2}{n}}\right)}=? \\ $$ Commented by Henri Boucatchou last updated on 17/Apr/20 $$\boldsymbol{{Check}}\:\:\boldsymbol{{once}}\:\:\boldsymbol{{more}}\:\:\boldsymbol{{Sir}},\:\:\:\boldsymbol{{ln}}\underset{{i}}…

lim-x-1-1-sin-1-x-1-x-e-1-1-x-x-x-

Question Number 154776 by john_santu last updated on 21/Sep/21 $$\:\underset{{x}\rightarrow\infty} {\mathrm{lim}}\left(\mathrm{1}+\frac{\mathrm{1}−\frac{\mathrm{sin}\:\left(\frac{\mathrm{1}}{{x}}\right)}{\frac{\mathrm{1}}{{x}}}}{{e}−\left(\mathrm{1}+\frac{\mathrm{1}}{{x}}\right)^{{x}} }\right)^{{x}} =? \\ $$ Commented by mathdanisur last updated on 21/Sep/21 $$\mathrm{say}\:\:\frac{\mathrm{1}}{\mathrm{x}}\:=\:\mathrm{t}\:\Rightarrow\:\mathrm{t}\rightarrow\mathrm{0} \\ $$$$\mathrm{e}^{\underset{\boldsymbol{\mathrm{t}}\rightarrow\mathrm{0}}…

Question-23688

Question Number 23688 by A1B1C1D1 last updated on 04/Nov/17 Answered by $@ty@m last updated on 04/Nov/17 $${Use}\:{method}\:{of}\:{rationalization} \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\:\frac{{x}^{\mathrm{2}} \left(\sqrt{{x}^{\mathrm{2}} +\mathrm{12}}+\sqrt{\mathrm{12}}\right)}{{x}^{\mathrm{2}} +\mathrm{12}−\mathrm{12}} \\ $$$$=\underset{{x}\rightarrow\mathrm{0}}…