Proof for $e^z = \lim \limits_{x \rightarrow \infty} \left( 1 + \frac{z}{x} \right)^x$
Solution 1:
Let $y=x/z$. Then $$ \lim \limits_{x \rightarrow \infty} \left( 1 + \frac{z}{x} \right)^x = \lim \limits_{y \rightarrow \infty} \left( 1 + \frac{1}{y} \right)^{yz} = \left(\lim \limits_{y \rightarrow \infty} \left( 1 + \frac{1}{y} \right)^{y}\right)^z = e^z $$
But note that this assumes that you have a definition of $a^x$ and know that $a^{xy}=(a^x)^y$ and that $x\mapsto a^x$ is continuous.