What i call the second Euler's Number

[x2thay]

Everyone of us know about the famous euler's number, which is e, which is aproximately 2.7182818...(as far as i cam remember)...which is used for many things in calculus...

Well, i was wondering where the heck does e come from and i realize after searching in the web, that e is the limit as x aproaches to the infinite, of the function f(x)=(1+1/x)^x or f(x)=(1+x)^(1/x). And this is awesome! Because i had no idea e was a limit! lol

Anyway, next thing i was wondering about was: what if the '+' sinal in the function f(x)=(1+1/x)^x gets switched to an '-'? The function would become: f(x)=(1-1/x)^x.

Here are the functions: http://www.geocities.com/just_dre/e3e.GIF

Well, i graphed both functions and as expected, f(x)=(1+1/x)^x aproachs to e. But the other function (the grey one) has a different limit which is aproximately 0,3678794409875026009331610590813...

My question is: does this new irrational constant have any meaning? If f(x)=(1+1/x)^x has, why can't f(x)=(1-1/x)^x?

Hope you can help me...^^

iMiguel

 

[quasar987]

it's fairly easy to see that for any real number y,

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

In particular,

$$\lim_{x\rightarrow \infty}\left(1-\frac{1}{x}\right)^x = e^{-1}$$

Note also that e^y can equivalently be defined in terms of its power series expansion as the solution to the differential equation f '(y)=f(y).

 

[a.a]

The number is defined as the limit of (1+1/n)^n as n approched infinity.
My question is why is that limit the number e. If you evaluate it useing the subsistutaion method, you get the limit is 1.

E is also defined as the limit of (1+n)^(1/n) as n approaches 0. Using the substitution method isn't this limit also 1?

 

[CompuChip]

The number is defined as the limit of (1+1/n)^n as n approched infinity.
My question is why is that limit the number e.

One could take that as a definition. Or one could prove that it satisfies the definition you take for e, for example, if you define e to be the number such that $de^y/dy = e^y$ you can differentiate the expression in quasars post and show that it equals its own derivative in the limit $x \to \infty$, or you can plug in the power expansion and show that it is the same number.

If you evaluate it useing the subsistutaion method, you get the limit is 1. E is also defined as the limit of (1+n)^(1/n) as n approaches 0. Using the substitution method isn't this limit also 1?

You made me curious, can you show us how you do that?

 

[Hurkyl]

If you evaluate it useing the subsistutaion method, you get the limit is 1.

No you don't; $1^{+\infty}$ is an indeterminate form.