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Mathematics
Calculus
Naturals or Reals When Taking Limits to Obtain the Value of Euler's Number e?
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[QUOTE="mcastillo356, post: 6826089, member: 506793"] [B]TL;DR Summary:[/B] 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: [B]Definition: [/B]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}## [B]Questions: [/B] 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)?. [B]Attempt: [/B]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}}##. [/QUOTE]
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Mathematics
Calculus
Naturals or Reals When Taking Limits to Obtain the Value of Euler's Number e?
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