How do you find the limit of (1-(2/x))^x as x approaches infinity?

Oct 20, 2017

I would rewrite to use ${\lim}_{t \rightarrow \infty} {\left(1 + \frac{1}{t}\right)}^{t} = e$ and continuity of the exponential function.

Explanation:

${\left(1 - \frac{2}{x}\right)}^{x} = {\left(1 - \frac{1}{\frac{x}{2}}\right)}^{x} = {\left({\left(1 - \frac{1}{\frac{x}{2}}\right)}^{\frac{x}{2}}\right)}^{2}$

As $x \rightarrow 00$, we have $\frac{x}{2} \rightarrow \infty$, so ${\left({\left(1 - \frac{1}{\frac{x}{2}}\right)}^{\frac{x}{2}}\right)}^{2} \rightarrow {e}^{2}$