Asymptotic expansion of $e^x$ other than Taylor series

Question

Are there any asymptotic expansions of $e^x$ as $x$ goes to 0 other than Taylor series?

Answer 1

The most elemantary ones are maybe $$1 + x \leq e^x \leq \frac{1}{1-x}$$ for $|x| < 1$. (The lower bound holds for all $x \in \mathbb{R}$.) These can be refined by using $e^x = e^{(x/n) \cdot n}$ into $$\left(1+\frac{x}{n} \right)^n \leq e^x \leq \left( 1 - \frac{x}{n} \right)^{-n} = \left( 1 + \frac{x}{n - x} \right)^n$$ for $n \geq 1$ and $|x| < n$.