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

Question

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):

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

http://math.stackexchange.com/questions/294855/deriving-eix-1-ix-fracix22 — lab bhattacharjee, Feb 05 '13 at 17:29

score 4 · Accepted Answer · answered Feb 05 '13 at 17:34

4

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.

answered Feb 05 '13 at 17:34

lhf

216,483

1

And note that $z$ is considered as constant while $x$ varies, tending to $\infty$, hence $x/z\to\infty$, too. – Berci Feb 05 '13 at 17:51
Nicely shown but I still need anwser to Q1. – 71GA Feb 05 '13 at 18:46
@lhf, how do you get y -> inf here? If z is negative, won't y -> -inf when x -> inf? – nullptr Mar 15 '24 at 23:48

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

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