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lim-x-x-ln-x-2-1-1-e-x-3-




Question Number 202466 by Rydel last updated on 27/Dec/23
lim_(x→+∞) ((x(√(ln (x^2 +1))))/(1+e^(x−3) ))
$$\underset{{x}\rightarrow+\infty} {\mathrm{lim}}\frac{{x}\sqrt{\mathrm{ln}\:\left({x}^{\mathrm{2}} +\mathrm{1}\right)}}{\mathrm{1}+{e}^{{x}−\mathrm{3}} } \\ $$
Answered by Mathspace last updated on 27/Dec/23
=lim_(x→+∞) xe^(−x+3) (√(ln(1+x^2 )))/1+e^(−x+3)   =lim_(x→+∞) xe^(−x+3) (√(2lnx+ln(1+(1/x^2 ))))  =lim_(x→+∞) xe^(−x+3) (√(2lnx))  =0  the boss here is e^(−x)
$$={lim}_{{x}\rightarrow+\infty} {xe}^{−{x}+\mathrm{3}} \sqrt{{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}/\mathrm{1}+{e}^{−{x}+\mathrm{3}} \\ $$$$={lim}_{{x}\rightarrow+\infty} {xe}^{−{x}+\mathrm{3}} \sqrt{\mathrm{2}{lnx}+{ln}\left(\mathrm{1}+\frac{\mathrm{1}}{{x}^{\mathrm{2}} }\right)} \\ $$$$={lim}_{{x}\rightarrow+\infty} {xe}^{−{x}+\mathrm{3}} \sqrt{\mathrm{2}{lnx}} \\ $$$$=\mathrm{0}\:\:{the}\:{boss}\:{here}\:{is}\:{e}^{−{x}} \\ $$

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