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 58239 by salahahmed last updated on 20/Apr/19

lim_(x→0^+ )  (x^(1/x) )

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

Answered by salahahmed last updated on 23/Apr/19

lim_(x→0^+ )  (e^((ln(x))/x) )  e^(lim_(x→0^+ )  (((ln(x))/x)))   e^(lim_(x→0^+ )  (ln(x)×(1/x)))   e^(ln(0^+ )×(1/0^+ ))   e^(∞(−∞))   e^(−∞) =0  ∴ lim_(x→0^+ )  (x^(1/x) )=0

$$\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\left({e}^{\frac{\mathrm{ln}\left({x}\right)}{{x}}} \right) \\ $$$${e}^{\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\left(\frac{\mathrm{ln}\left({x}\right)}{{x}}\right)} \\ $$$${e}^{\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\left(\mathrm{ln}\left({x}\right)×\frac{\mathrm{1}}{{x}}\right)} \\ $$$${e}^{\mathrm{ln}\left(\mathrm{0}^{+} \right)×\frac{\mathrm{1}}{\mathrm{0}^{+} }} \\ $$$${e}^{\infty\left(−\infty\right)} \\ $$$${e}^{−\infty} =\mathrm{0} \\ $$$$\therefore\:\underset{{x}\rightarrow\mathrm{0}^{+} } {\mathrm{lim}}\:\left({x}^{\frac{\mathrm{1}}{{x}}} \right)=\mathrm{0} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com