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Question Number 72905 by Tony Lin last updated on 04/Nov/19

$$f\left(x\right)⩾0,and\underset{x\to a}{\lim }f\left(x\right)=0,\underset{x\to a}{\lim }g\left(x\right)=\mathrm{\infty }$$$$then\underset{x\to a}{\lim }f{\left(x\right)}^{g\left(x\right)}=?$$

Commented by mathmax by abdo last updated on 04/Nov/19

$$wehavebytaylorf\left(x\right)=f\left(a\right)+(x-a)f^{{}^{\prime }}\left(a\right)+o(x-a)$$$$\Rightarrow f\left(x\right)\sim (x-a)f^{{}^{\prime }}\left(a\right)because{lim}_{x\to a}f\left(x\right)=0$$$$\left(wesupposefderivableatpointa\right)\Rightarrow$$$$f{\left(x\right)}^{g\left(x\right)}\sim {\left\{(x-a)f^{{}^{\prime }}\left(a\right)\right\}}^{g\left(x\right)}=e^{g\left(x\right)ln\left\{(x-a)f^{{}^{\prime }}\left(a\right)\right\}}$$$$if{lim}_{x\to a}g\left(x\right)=+\mathrm{\infty }wehave{lim}_{x\to a^{+}}ln\left\{(x-a)f^{{}^{\prime }}\left(a\right)\right\}=-\mathrm{\infty }\Rightarrow$$$${lim}_{x\to a^{+}}g\left(x\right)ln\left\{(x-a)f^{{}^{\prime }}\left(a\right)\right\}=-\mathrm{\infty }\Rightarrow$$$${lim}_{x\to a^{+}}{\left\{f\left(x\right)\right\}}^{g\left(x\right)}=0$$$$if{lim}_{x\to a^{+}}g\left(x\right)=-\mathrm{\infty }\Rightarrow {lim}_{x\to a^{+}}g\left(x\right)ln\left\{(x-a)f^{{}^{\prime }}\left(a\right)\right\}=+\mathrm{\infty }\Rightarrow$$$${lim}_{x\to a^{+}}{\left\{f\left(x\right)\right\}}^{g\left(x\right)}=+\mathrm{\infty }andifyouhaveaclearanswerpostit..$$

Commented by Tony Lin last updated on 05/Nov/19

$$thankssir$$

Commented by mathmax by abdo last updated on 05/Nov/19

$$youarewelcome.$$

Answered by mind is power last updated on 04/Nov/19

$$\mathrm{f}{\left(\mathrm{x}\right)}^{\mathrm{g}\left(\mathrm{x}\right)}={\mathrm{e}}^{\mathrm{g}\left(\mathrm{x}\right)\ln \left(\mathrm{f}\left(\mathrm{x}\right)\right)}$$$$\lim \mathrm{g}\left(\mathrm{x}\right)=+\mathrm{\infty }$$$$\lim [\mathrm{f}\left(\mathrm{x}\right)\to 0$$$$\Rightarrow \lim \ln \left(\mathrm{f}\left(\mathrm{x}\right)\right)\to -\mathrm{\infty }$$$$\Rightarrow \mathrm{g}\left(\mathrm{x}\right)\ln \left(\mathrm{f}\right)\to -\mathrm{\infty }$$$$\Rightarrow {\mathrm{e}}^{\mathrm{g}\left(\mathrm{x}\right)\ln \left(\mathrm{f}\left(\mathrm{x}\right)\right)}\to 0$$

Commented by Tony Lin last updated on 05/Nov/19

$$thankssir$$

Answered by Tony Lin last updated on 05/Nov/19

$$\underset{x\to a}{\lim }f{\left(x\right)}^{g\left(x\right)}=0$$

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