Question

how can we evaluate lim>infinity (1-2/x)^(x+1)

Answer 1

It is known that
\[
\lim_{x\to \infty}\left (1 + \frac a x\right)^x=e^a
\]
In your case a=2, so the the limit is \(e^{-2}\)

Answer 2

To prove the above statement take the log of the expression and find its limit,