Limit of $\left(1 - \frac{1}{x}\right)^{x}$ as $x \to \infty$

The limit of ≤ft(1 - (1)/(x))^x as x → is e^-1.

The limit of $\left(1 - \frac{1}{x}\right)^{x}$ as $x \to \infty$ is $e^{-1}$ .

\lim_{x \to \infty} \left(1 - \frac{1}{x}\right)^{x} = e^{-1}

Graph of (1 - 1 / x)^x near x = ∞. The curve levels off at 1/e as x grows.

How to work it out

1
We want what (1 - 1 / x)^x approaches as x → ∞.
$\lim_{x \to \infty} \left(1 - \frac{1}{x}\right)^{x}$
2
Direct substitution gives an indeterminate form: 1^∞. It does not tell us the limit.
3
Method: replace 1 by −1 in (1 + a/n)ⁿ → eᵃ.
4
Result:
$\lim_{x \to \infty} \left(1 - \frac{1}{x}\right)^{x} = e^{-1}$

LaTeX source of the formulas on this page

\[ \lim_{x \to \infty} \left(1 - \frac{1}{x}\right)^{x} = e^{-1} \]
\[ \lim_{x \to \infty} \left(1 - \frac{1}{x}\right)^{x} \]

What is the limit of (1 - 1 / x)^x as x approaches ∞?

The limit of $\left(1 - \frac{1}{x}\right)^{x}$ as $x \to \infty$ is $e^{-1}$ .

Which form is it?

1^∞.