Answer by WHLin for How do we show that $\lim_{x\to0}(1+x)^{\frac{1}{x}}$ is a finite number

Let $f(x)\equiv (1+x)^{1/x}$, we have $f(x)=exp\{log[(1+x)^{1/x}]\}$. Since $exp\{.\}$ is continuous, we have

$$\lim_{x\rightarrow 0} f(x)= exp\{\lim_{x\rightarrow 0} log[(1+x)^{1/x}]\}= exp\{\lim_{x\rightarrow 0} \frac{log(1+x)}{x}\}$$

Note that $log(1+x) = x - \frac{x^2}{2}+[\textit{higher order terms}]$, and hence the limit inside the large bracket is 1. This concludes $\lim_{x\rightarrow 0} f(x) = e \approx 2.718$.