Prove $\sum\limits_{n=0}^\infty\frac{x^n}{n!}=\lim\limits_{n\rightarrow\infty}\left(1+\frac{x}{n}\right)^n$ [duplicate]

$$x\in\mathbb{R}$$. I can prove that both sides converge, and maybe we should show $$\limsup\limits_{n\rightarrow\infty}\left|\sum\limits_{k=0}^{n}\frac{x^k}{n!}-\left(1+\frac{x}{n}\right)^n \right|<\varepsilon$$, but the construction of the right hand side is a little tricky for me.

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• Show that they both solve the differential equation $f'(x) = f(x)$ with initial condition $f(0) = 1$ (the solution is unique; this isn't hard to prove). The LHS solves it by finding its Taylor series and the RHS solves it using Euler's method. – Qiaochu Yuan Nov 18 '18 at 22:15
• Expand the right side (binomial expansion) and match up term by term (powers of $x$) to the left side. – herb steinberg Nov 18 '18 at 22:24

(1) Define $$\exp(x)$$ to be the unique solution to $$y' = y$$ with $$y(0) = 1$$. Let $$\log$$ denote its inverse (which exists because $$\exp$$ is increasing).
(2) Use uniqueness of $$\exp$$ to show that $$\exp(x + y) = \exp(x) \exp(y)$$ and thus $$\log(xy) = \log(x) + \log(y)$$.
(3) Use the defining differential equation of $$\exp$$ to show that $$\exp(x) = \sum x^n/n!$$.
(4) Use the definition of $$\log$$ to show that $$\log'(1) = 1$$, then compute the log of the right-hand limit with l'Hopital's rule.