Talk:Exp x equals e^x

Sorry, but I don't see the point to this page. Euler's number is defined as that number which is such that $e^x = exp x$ in the general case, and the proof that the (1+1/n)^n expression gives e is done on another page. So demonstrating that elementary algebraic manipulation on just integers seems a bit limiting.

However, the expression:
 * $\displaystyle \lim_{n \to +\infty}\left({1 + \frac x n}\right)^n = \lim_{\left({n/x}\right) \to +\infty}\left (\left ({1 + \frac 1 {\left({n/x}\right)}} \right)^{\left({n/x}\right)}\right)^x$

looks quite neat, and might be usable in the page Definition:Exponential. But it would need to be made rigorous and demonstrated to apply to the domain of real numbers, not just $\N$. --prime mover 14:26, 12 January 2012 (EST)
 * Well I was thinking more of the first definition given here: Definition:Euler's Number, not the third or fourth. That's what I thought would be the point of the page, just another perspective on the consistency of definitions.
 * As for non-$\N$ indices, it can be done, but I would have to leave it as a . I think it's possible to do what you're asking me to do because I had a correspondence with dr math, I'll quote a part of it. --GFauxPas 14:48, 12 January 2012 (EST)

...so that $a^{1/2}$ should equal $\sqrt a$. Similarly, you can define $a^{m/n}$ as $n$th root of $a^m$ or ($n$th root of $a$)^$m$, but defining $a^\pi$ is still a challenge. You could define it through a limiting process: if $r_n$ is a sequence of rational numbers which converge to $\pi$, then you would hope that $\lim_{n \to \infty} a^{r_n} = a^{\pi}$, but there's a lot of work involved in proving this, showing that the result is independent of the particular sequence of rationals used, etc. Even after you have shown that this definition makes sense, you still have to show that the function $f(x)=a^x$ is continuous and then differentiable. Showing differentiability involves showing that the limit $\lim_{h \to 0} \frac {a^h - 1} h$ exists and determining its value. So there is a lot of work involved in verifying all the properties you want in exponential functions if you start from scratch. - Dr Fenton