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How these to definitions of $e^x$ equivalent?

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

I just can't prove that they are the same, so how do you do it?

ZYX
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DefinitelyNotADolphin
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    Substitute $n \to n/x$. – RDK Nov 16 '24 at 14:22
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    This question is similar to: prove the definitions of e to be equivalent. If you believe it’s different, please edit the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. – Kraken Nov 16 '24 at 14:22
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    You simply cannot write $$(1+\frac{1}{n})^{x \ n}$$ for integer n and real or complex $x$ without having defined real positive $a^b$ first. That implies the definition of $e^x$ and its inverse $\log x$ – Roland F Nov 16 '24 at 14:34
  • @Roland: $a^b$ can be defined as the limit of $a^{b_n}$ where $(b_n)$ is any sequence of rationals that tends to $b$. You dont need $e$ at all. – TonyK Nov 16 '24 at 16:51
  • Existence is easy, but you need the value and summation formula of exponents; possible to do but not recommended. – Roland F Nov 16 '24 at 18:54

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