Question and Answers Forum

All Questions      Topic List

Limits Questions

Previous in All Question      Next in All Question      

Previous in Limits      Next in Limits      

Question Number 91020 by jagoll last updated on 27/Apr/20

lim_(x→0)  (sin x)^(1/x)  ?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{sin}\:{x}\right)^{\frac{\mathrm{1}}{{x}}} \:?\: \\ $$

Commented by jagoll last updated on 28/Apr/20

it does mean lim_(x→0)  (sin x)^(1/x)  DNE?

$${it}\:{does}\:{mean}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{sin}\:{x}\right)^{\frac{\mathrm{1}}{{x}}} \:{DNE}? \\ $$

Commented by mathmax by abdo last updated on 27/Apr/20

let f(x)=(sinx)^(1/x)  ⇒f(x) =e^((1/x)ln(sinx))   ((ln(sinx))/x) ∼((ln(x))/x) →−∞  (x→0^+ )⇒lim_(x→0^+ )   f(x) =0

$${let}\:{f}\left({x}\right)=\left({sinx}\right)^{\frac{\mathrm{1}}{{x}}} \:\Rightarrow{f}\left({x}\right)\:={e}^{\frac{\mathrm{1}}{{x}}{ln}\left({sinx}\right)} \\ $$$$\frac{{ln}\left({sinx}\right)}{{x}}\:\sim\frac{{ln}\left({x}\right)}{{x}}\:\rightarrow−\infty\:\:\left({x}\rightarrow\mathrm{0}^{+} \right)\Rightarrow{lim}_{{x}\rightarrow\mathrm{0}^{+} } \:\:{f}\left({x}\right)\:=\mathrm{0} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com