How to calculate this limit without L'Hopital's rule

I know that in the very small values of $x$
$$(1+x)^n ≈1+ nx$$
and I can prove it using Taylor series.
But I wanted to prove it without any smell of derivative.

So... In order to calculate the following limit

$$\lim _{x \rightarrow\ 0} \frac{(1+x)^n-1} x,$$

I know that the result must be equal to $n$.

But is there any method without using l'Hospital's rule?

Note : n can be any real value not just for integers, so I didn't want to use binomial theorem.