Q:

Definition of e, how to relate that to other interest rates

I understand that one way of understanding the meaning of the number $e$ is to form a compound interest formula, $A = \left(1+\frac{1}{n}\right)^{nx}$ and then let $n\rightarrow \infty$ for which this converges to $e$.  However, this occurs by setting the principle to 1, the rate to 1, and the time to $x$.  And I see how you can manipulate this to get any other principle--you just multiply by the principle you want.  So $Pe^{x}$ would be any other principle.  But how do you get any other rate?  
That is to say, suppose you have $A = \left(1+\frac{r}{n}\right)^{nx}$ and let $n\rightarrow \infty$.  To what does this converge?

A:

It converges to $$e^{rx}$$
To see this you can write $$A = \left(1+\frac{r}{n}\right)^{nx} = \left(1+\frac{1}{\frac{n}{r}}\right)^{\frac{n}{r}xr} \to e^{xr}$$

