Can someone suggest me a quick manual method to find
$\lim_{x\rightarrow \infty}(\frac{x}{e}-x(\frac{x}{1+x})^x)$ ?
I'm just going on and on...
P.S:I'm just in high school..keep it down to my level
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1This is $(x/e)(1-e^{u})$ with $u=1-x\log(1+1/x)=1-x(1/x-1/(2x^2)+o(1/x^2))=1/(2x)+o(1/x)$ hence this is $(x/e)(1-e^{1/(2x)+o(1/x)})=(x/e)(-1/(2x)+o(1/x))\to-1/(2e)$. Tools: $\log(1+t)=t-t^2/2+o(t^2)$ and $e^t=1+t+o(t)$ when $t\to0$. – Did Nov 10 '15 at 16:00
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@Did Great ;-D!!make it an answer! – Nov 10 '15 at 16:06