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lim-x-0-sin-x-1-ln-x-




Question Number 99968 by bobhans last updated on 24/Jun/20
lim_(x→0^+ )  (sin x)^(1/(ln(x)))  =?
$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\left(\mathrm{sin}\:\mathrm{x}\right)^{\frac{\mathrm{1}}{\mathrm{ln}\left(\mathrm{x}\right)}} \:=?\: \\ $$
Commented by bobhans last updated on 24/Jun/20
=e^(lim_(x→0)  ((ln(sin x))/(ln(x)))) = e^(lim_(x→0)  ((((cos x)/(sin x))/(1/x))))  = e^(lim_(x→0)  ((xcos x)/(sin x))) = e
$$=\mathrm{e}^{\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{ln}\left(\mathrm{sin}\:\mathrm{x}\right)}{\mathrm{ln}\left(\mathrm{x}\right)}} =\:\mathrm{e}^{\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\frac{\frac{\mathrm{cos}\:\mathrm{x}}{\mathrm{sin}\:\mathrm{x}}}{\frac{\mathrm{1}}{\mathrm{x}}}\right)} \:=\:\mathrm{e}^{\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{xcos}\:\mathrm{x}}{\mathrm{sin}\:\mathrm{x}}} =\:\mathrm{e} \\ $$

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