- #1

mcastillo356

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- TL;DR Summary
- I've found two ways of taking limits in order to obtain ##e## number. Do they belong to naturals in one case and to the reals in the second case?

Hi PF

Searching on the Internet, I've found this definition:

(i) ##e=\displaystyle\lim_{x\to{0}}{(1+x)^{\displaystyle\frac{1}{x}}}##

and

(ii) ##e=\displaystyle\lim_{n\to{\infty}}{(1+\displaystyle\frac{1}{n})^n}##

1-Does it make sense Definition (i)? I don't think so: ##(1+x)## on the base is strange: a lonely ##x## on the sum.

2-If both make sense, does ##x\in{\mathbb{R}}## in (i), and ##n\in{\mathbb{N}}## in (ii)?.

Searching on the Internet, I've found this definition:

**Definition:**Euler's Number as a Limit(i) ##e=\displaystyle\lim_{x\to{0}}{(1+x)^{\displaystyle\frac{1}{x}}}##

and

(ii) ##e=\displaystyle\lim_{n\to{\infty}}{(1+\displaystyle\frac{1}{n})^n}##

**Questions:**1-Does it make sense Definition (i)? I don't think so: ##(1+x)## on the base is strange: a lonely ##x## on the sum.

2-If both make sense, does ##x\in{\mathbb{R}}## in (i), and ##n\in{\mathbb{N}}## in (ii)?.

**Attempt:**I think I've fallen into an erratic web; but let's suppose the contrary. In that case, no matters reals neither naturals. I make no distinction. It could be also ##x\in{\mathbb{N}}##, and ##n\in{\mathbb{R}}##.