Proposition: Infinitesimal Exponential Growth is the Growth of the Identity Function

The exponential function growth at $0$ is the same as the growth of the identify function, formally $$ \lim_{x\to 0,~x\neq 0}\frac{e^x-1}{x}=1.$$

The following figure visualizes the behavior of both functions at $0$ (for a better comparison, instead of the identity function $f(x)=x$, the function $f(x)=x+1$ was drawn together with the function $g(x)=\exp(x)$:

Proofs: 1

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983