What is the value of $\lim\limits_{n \to \infty}\left(1+\dfrac{x}{n}\right)^n$ (with answer)

$\lim\limits_{n \to \infty}\left(1+\dfrac{x}{n}\right)^n = e^x$ since:

(1)  For $x = 0$, $\lim\limits_{n \to \infty}\left(1+\dfrac{0}{n}\right)^n =  1 = e^0$

(2)  So we can assume that $x \ne 0$ so that:

$$\lim\limits_{n \to \infty}\left(1+\dfrac{x}{n}\right)^n = \lim\limits_{n \to \infty}e^{n\ln\left(1+\dfrac{x}{n}\right)} = \lim\limits_{n \to \infty}e^{x\ln\left(\frac{1+x/n}{x/n}\right)}$$

(3)  Since $\lim\limits_{n \to \infty}\dfrac{1+x/n}{x/n} = \lim\limits_{h \to 0}\dfrac{1+h}{h}$:

$$\lim\limits_{n \to \infty}e^{x\ln\left(\frac{1+x/n}{x/n}\right)} = \lim\limits_{h \to 0}e^{x\ln\left(\frac{1+h}{h}\right)} = e^{x\lim\limits_{h \to 0}\left(\ln\left[\frac{1+h}{h}\right]\right)}$$

(4)  Since $\lim\limits_{h \to 0}\dfrac{\text{ln}(1+h)}{h}=1$ (see here if needed):

$$e^{x\lim\limits_{h \to 0}\left(\ln\left[\frac{1+h}{h}\right]\right)} = e^x$$