# Proof for $e^z = \lim \limits_{x \rightarrow \infty} \left( 1 + \frac{z}{x} \right)^x$

Q1: Could someone provide a proof for this equation (please focus on this question):

$$e^z = \lim \limits_{x \rightarrow \infty} \left( 1 + \frac{z}{x} \right)^x$$

Q2: Is there any corelationn between above equation and the equation below (this would benefit me a lot because i at least know how to proove this one):

$$e = \lim \limits_{x \rightarrow \infty} \left( 1 + \frac{1}{x} \right)^x$$

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For $Q2$, set $z=1$, $e^1 = e$ – Stefan Feb 5 '13 at 17:31
Thank you. How about proof for Q1? – 71GA Feb 5 '13 at 17:37

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.
And note that $z$ is considered as constant while $x$ varies, tending to $\infty$, hence $x/z\to\infty$, too. – Berci Feb 5 '13 at 17:51