I've seen in many textbooks that the following is just a definition: $$e^x = \sum_{n=0}^{\infty} \frac{x^{n}}{n!}$$
And then many textbooks just go ahead to prove the absolute convergence of the infinite series using, for example, ratio test, to conclude that the infinite sum actually makes sense. But even if the infinite series is absolutely convergent, how do I know the series does converge to $e^x$ instead of some other functions? The series convergence tests never mention about the limit that the series converges to. How do people come up with such a definition at the first place?