Limit that gives a power of e

Statement

Calculate $\displaystyle\lim_{x\to+\infty}\left(1+\frac{2}{x}\right)^x$.

Solution

A $1^\infty$ form. We reduce to the standard limit $\left(1+\frac{1}{t}\right)^t\to e$ with the substitution $t = \dfrac{x}{2}$ (so $\dfrac{2}{x}=\dfrac{1}{t}$ and $x = 2t$, with $t\to+\infty$): $\left(1+\frac{2}{x}\right)^x = \left(1+\frac{1}{t}\right)^{2t} = \left[\left(1+\frac{1}{t}\right)^{t}\right]^{2} \to e^2.$ Hence the limit is $\boxed{e^2}$.

Alternatively, with the trick $A^B=e^{B\ln A}$: the exponent is $x\ln\!\left(1+\frac{2}{x}\right)\sim x\cdot\frac{2}{x}=2$, whence $e^2$.

Links

Topics: Limits
Concepts: Indeterminate forms · Standard limits
Skills: Calculating limits
Exercise type: Limit calculation