Menu Close

if-it-s-correct-solve-plz-lim-x-0-x-x-1-xlnx-




Question Number 12328 by Mr Chheang Chantria last updated on 19/Apr/17
if it′s correct solve plz  lim_(x→0) ((x^x −1)/(xlnx))
$$\boldsymbol{\mathrm{if}}\:\boldsymbol{\mathrm{it}}'\boldsymbol{\mathrm{s}}\:\boldsymbol{\mathrm{correct}}\:\boldsymbol{\mathrm{solve}}\:\boldsymbol{\mathrm{plz}} \\ $$$$\underset{\boldsymbol{\mathrm{x}}\rightarrow\mathrm{0}} {\boldsymbol{\mathrm{lim}}}\frac{\boldsymbol{\mathrm{x}}^{\boldsymbol{\mathrm{x}}} −\mathrm{1}}{\boldsymbol{\mathrm{xlnx}}} \\ $$
Answered by mrW1 last updated on 19/Apr/17
=lim_(x→0)  ((e^(xln x) −1)/(xln x))  =lim_(x→0)  ((x^x (1+ln x))/(1+ln x))  =lim_(x→0)  x^x   =1
$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{e}^{{x}\mathrm{ln}\:{x}} −\mathrm{1}}{{x}\mathrm{ln}\:{x}} \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{x}^{{x}} \left(\mathrm{1}+\mathrm{ln}\:{x}\right)}{\mathrm{1}+\mathrm{ln}\:{x}} \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}^{{x}} \\ $$$$=\mathrm{1} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *