Intuitively, why are the two limit definitions of $e^x$ equivalent? Thanks for reading!

Intuitively, why does...

$$\lim_{n \rightarrow \infty} \left(1+\frac{1}{n}\right)^{xn}=\lim_{n \rightarrow \infty} \left(1+\frac{x}{n}\right)^{n}=e^x$$

Note, I'm not asking why $e^x$ is one of the two limits. I understand the first limit, or at least I think I do.

In terms of continuous growth however (I don't just want a mathematical proof - the more intuitive the answer is the better), I'd like to understand why the two limits are equivalent!

Why is letting some principal amount grow by $\frac{1}{n}$ times its current value $xn$ times equal to letting that principal grow by $\frac{x}{n}$ times its current value $n$ times if we allow $n \rightarrow \infty$?

Thanks!
Forget that $n$ is an integer. For $x &gt; 0$ we have

$$\lim_{y\to\infty} \left(1+\frac{x}{y}\right)^y = \lim_{y\to\infty} \left(1+\frac{1}{\frac{y}{x}}\right)^{x\cdot \frac{y}{x}}$$

If $y \to \infty$ then $z:= \frac{y}{x} \to \infty$ as well so change of variables gives that this is equal to $$\lim_{z\to\infty} \left(1+\frac1z\right)^{xz}$$
which is the desired expression.
I will say my first thought. 

If fix $x$,  one can take $\exp (x)$ for "constant" intuitively. And 
 it is known that for a real number $x'$ there are infinite sequences convergent to it, I mean there are many many sequences we do not construct (or notice) with the same limit $x'$, 
so let $a_n = (1+\frac{x}{n})^n$, $b_n = (1+\frac{1}{n})^{xn}$, sequences $\{ a_n \} ,  \{ b_n \}$  are just two cases( we are familiar with ) among them. 

Maybe there is nothing special comparing to the rest.