Limit of $\left(1 + \frac{2}{x}\right)^{x}$ as $x \to \infty$

The limit of ≤ft(1 + (2)/(x))^x as x → is e^2.

The limit of $\left(1 + \frac{2}{x}\right)^{x}$ as $x \to \infty$ is $e^{2}$ .

\lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^{x} = e^{2}

Graph of (1 + 2 / x)^x near x = ∞. The curve levels off at e² as x grows.

How to work it out

1
We want what (1 + 2 / x)^x approaches as x → ∞.
$\lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^{x}$
2
Direct substitution gives an indeterminate form: 1^∞. It does not tell us the limit.
3
Method: write (1 + 2/n)ⁿ = [(1 + 1/m)ᵐ]² with m = n/2.
4
Result:
$\lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^{x} = e^{2}$

LaTeX source of the formulas on this page

\[ \lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^{x} = e^{2} \]
\[ \lim_{x \to \infty} \left(1 + \frac{2}{x}\right)^{x} \]

What is the limit of (1 + 2 / x)^x as x approaches ∞?

The limit of $\left(1 + \frac{2}{x}\right)^{x}$ as $x \to \infty$ is $e^{2}$ .

Which form is it?

1^∞.