Define $e(x)=\lim_{n\to\infty}\left(1+x/n \right)^{n}$
Define $l(x)=e^{-1}(x)$ (since $e$ can be shown to be invertible)
Define $A(a,x)=e(x\cdot l(a)),a>0$
Prove that the following limit exists and find its value:
$$\lim_{x\to x_0}\frac{A(a,x)-A(a,x_0)}{x-x_0},a>0$$
The issue that I have with this problem is that I need to be able to prove the limit WITHOUT yet having proved that $A(a,x)=a^x$ is true $\forall a\in\mathbb{R}\ge0$. So I'm honestly not sure how to go about this from here. Without being able to write it that way, how can I make any progress with this?
So I would rewrite $$\lim_{x \rightarrow x_0} \frac{A(a,x)-A(a,x_0)}{x-x_0}$$ as $$\lim_{x \rightarrow x_0} \frac{e^{x\log a}-e^{x_0\log a}}{x-x_0} = e^{x_0\log a}\log a \lim_{h \rightarrow 0} \frac{e^{h\log a}-1}{h\log a} = A(a,x_0)\log a.$$