Question and Answers Forum

All Questions      Topic List

Vector Calculus Questions

Previous in All Question      Next in All Question      

Previous in Vector Calculus      Next in Vector Calculus      

Question Number 64011 by Prithwish sen last updated on 12/Jul/19

$$\mathrm{li}\underset{\mathrm{x}\to 0}{\mathrm{m}}\frac{{\mathrm{x}}^{\mathrm{x}}-1}{\mathrm{xlnx}}$$

Commented by kaivan.ahmadi last updated on 12/Jul/19

$$y=x^x\Rightarrow lny=xlnx\Rightarrow \frac{y^{\prime }}{y}=lnx+x\times \frac{1}{x}=1+lnx\Rightarrow$$$$y^{\prime }=y(1+lnx)=x^x(1+lnx)$$$${lim}_{x\to 0^{+}}\frac{x^x-1}{xlnx}={lim}_{x\to 0^{+}}\frac{x^x-1}{{lnx}^x}\stackrel{hop}{=}$$$${lim}_{x\to 0^{+}}\frac{x^x(1+lnx)}{\frac{x^x(1+lnx)}{x^x}}={lim}_{x\to 0^{+}}{x}^x=$$$${lim}_{x\to 0^{+}}{e}^{xlnx}=e^0=1$$

Commented by mathmax by abdo last updated on 12/Jul/19

$$letA\left(x\right)=\frac{x^x-1}{xlnx}\Rightarrow A\left(x\right)=\frac{e^{xln\left(x\right)}-1}{xlnx}wehave{lim}_{x\to 0^{+}}xln\left(x\right)=0$$$$and{e}^u\sim 1+u(u\in V\left(0\right))\Rightarrow e^{xln\left(x\right)}=1+xln\left(x\right)\Rightarrow$$$$e^{xln\left(x\right)}-1\sim xln\left(x\right)(x\to 0)\Rightarrow \frac{e^{xln\left(x\right)}-1}{xlnx}\sim 1\Rightarrow$$$${lim}_{x\to 0^{+}}A\left(x\right)=1.$$

Commented by Prithwish sen last updated on 12/Jul/19

$$\mathrm{thank}\mathrm{you}\mathrm{sirs}.$$

Answered by Rio Michael last updated on 12/Jul/19

$$(lnx+1)x^x$$$$lety=x^{x}since\frac{d(-1)}{dx}=0$$$$lny={lnx}^x$$$$\frac{1}{y}=x\left(\frac{1}{x}\right)+lnx$$$$\frac{1}{y}=lnx+1$$$$=(lnx+1)x^x$$$$\underset{x\to 0}{lim}\frac{(lnx+1)x^x}{lnx+1}$$$$\underset{x\to 0}{lim}{x}^x$$$$=0$$$$pleasecheck$$

Commented by Prithwish sen last updated on 12/Jul/19

$$\mathrm{thamk}\mathrm{you}\mathrm{sir}$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com