Finding exponential function limit definition from definition of e I am trying to prove, from the following limit definition of $e$
$$e=\lim_{n\to+\infty} \left(1+\frac{1}{n}\right)^n$$
the following definition for the exponential function:
$$e^x =\lim_{n\to+\infty} \left(1+\frac{x}{n}\right)^n$$
I tried to follow the answer of this post, but there is something I don't understand: Prove $e^x$ limit definition from limit definition of $e$.
Here is a screenshot:

To substitute $n$ for $u$ in the limit "index", we need to make sure that $u$ goes to $+\infty$ as $n$ goes to $+\infty$.
Therefore, we should have:
$$\lim_{n\to+\infty} u = \lim_{n\to+\infty} nx \stackrel{?}{=}+\infty$$
Which is, in some way, $(+∞)*x$.
That is fine if $x$ is positive, but what if $x$ were negative? Wouldn't that limit then be equal to $(-∞)$, which would invalidate that "re-indexing"?
Is the proof missing something, or am I wrong?
Thanks in advance!
(The reason I am writing this here and not commenting is that the site settings forbid me to, because of my limited activity.)

If you don't have any restrictions on how to prove, then you can use logarithm method-
Let $f(n) = (1 + \frac{1}{n})^{n}$. Then,
$\lim\limits_{n \to \infty} f(n) = \lim\limits_{n \to \infty} (1 + \frac{1}{n})^{n} = \Large e^{\big[\large \lim\limits_{n \to \infty} \large\ln \big(1 + \frac{1}{n}\big)^{n}\big]}$
So, instead of finding $\lim\limits_{n \to \infty} f(n)$, try to find $\lim\limits_{n \to \infty} \ln(f(n))$.
So, $\lim\limits_{n \to \infty} \ln(f(n)) = \lim\limits_{n \to \infty} \ln \big(1 + \frac{1}{n}\big)^{n}$
$= \lim\limits_{n \to \infty} n\ln \big(1 + \frac{1}{n}\big) = \lim\limits_{n \to \infty} \frac{\ln \big(1 + \frac{1}{n}\big)}{1/n}$
Now this is $\frac{0}{0}$ form, so using L'Hôpital's rule,
$\lim\limits_{n \to \infty} \ln(f(n)) = \lim\limits_{n \to \infty} \frac{1}{1+1/n}*\big(\frac{-1}{n^{2}}\big)*\left(\Large\frac{1}{\frac{\Large -1}{\Large n^{2}}}\right)$
$= \lim\limits_{n \to \infty} \frac{1}{1+1/n} = 1$
So, $\lim\limits_{n \to \infty} f(n) = e^{1} = e$
Same can be used for proving $\lim\limits_{n \to \infty} (1 + \frac{x}{n})^{n} = e^{x}$
