POTW Find Limit of $$\frac{x}{e} - \left(\frac{x}{x+1}\right)^x$$ at Infinity

[Euge]

Find the limit $$\lim_{x\to \infty} x\left[\frac{1}{e} - \left(\frac{x}{x+1}\right)^x\right]$$

 

[anuttarasammyak]

Spoiler

Substiuting x=1/y, the task is
\lim_{y\rightarrow +0}\frac{e^{-1}-(1+y)^{-1/y}}{y}
Considering
-\frac{1}{y}\ln(1+y) \approx -\frac{1}{y} (y+\frac{y^2}{2})
the task is
\lim_{y\rightarrow +0}\frac{e^{-1}(1-e^{-y/2})}{y}=(2e)^{-1}
[EDIT]
-\frac{1}{y}\ln(1+y) \approx -\frac{1}{y} (y-\frac{y^2}{2})
the task is
\lim_{y\rightarrow +0}\frac{e^{-1}(1-e^{y/2})}{y}=-(2e)^{-1}

 

[malawi_glenn]

@anuttarasammyak

Shouldn't it be -1/(2e) ?

 

[anuttarasammyak]

My bad, wrong sign in expansion of log.