limit of an expression

Question

What is the limit of this expression in terms of some known mathematical constants (if any)? $$\lim_{x\to\infty} \left(\frac{x-1}{x}\right)^x$$

Answer 1

$$y=\left(\frac{x-1}{x}\right)^x\implies \log(y)=x \log\left(\frac{x-1}{x}\right)=x \log\left(1-\frac{1}{x}\right)$$ Using Taylor series$$\log(y)=x\left(-\frac{1}{x}-\frac{1}{2 x^2}+O\left(\frac{1}{x^3}\right)\right)=-1-\frac{1}{2 x}+O\left(\frac{1}{x^2}\right)$$ Continuing with $$y=e^{\log(y)}=\frac{1}e\left(1-\frac{1}{2 x}\right)+O\left(\frac{1}{x^2}\right)$$ which shows the limit and also how it is approached.

Answer 2

I guess this can be expressed as $$ \lim(1 - 1/x)^x = e^{-1}. $$

But, I wonder if this question fits to this forum.