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.

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

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

Graph of (1 + 1 / x)^x near x = ∞. The curve levels off at 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: this is one definition of e (compound interest with ever more frequent compounding).
4
Result:
$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x} = e$

LaTeX source of the formulas on this page

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

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 $e$ .

Which form is it?

1^∞.