Question

How does the limit [1 + (1/x)]^x as
x -> ± ∞

equal "e"? <-- Euler's number..

Answer 1

Because it gets closer and closer to 2.718.

Answer 2

\[(1+(1/x))^x\]

Answer 3

actually, it is [1 + 1/x]^x

Answer 4

yea amistre is right, so how..

Answer 5

How can I convince myself that this is true through algebra or calculus?

Answer 6

Calculus. :)

Answer 7

Hang on a second: how can you take the limit of 1/x inside the bracket, when the whole thing is being raised to the power of "x"?

Answer 8

it is 3 more or less by definition

Answer 9

?

Answer 10

i meant e of course

Answer 11

OK I have my piece of paper, pencil and calculator out right now. Tell me how I should proceed with convincing myself that this is true.

Answer 12

\[e=\lim_{n\rightarrow \infty}(1+\frac{1}{n})^n\] is one definition

Answer 13

it depends on what your definition of "e" is. it it is the one i wrote, then there is nothing to prove.

Answer 14

still requires proof that the limit actually converges

Answer 15

http://www.wolframalpha.com/input/?i=lim+x-%3Einfinity+%281%2B1%2Fx%29^x

Answer 16

click show steps

Answer 17

really if you want to make sense out of it you need to say what e is. the wolfram steps are silly in this case

Answer 18

lol

Answer 19

but zarkon is right. what you need to show is that it converges to something. it does, and we call that something "e" so to say it converges to e is a tautology, unless you have some other definition of e

Answer 20

you also said an algebraic way
you can always see that the limit is getting close to e by taking large and larger values of x and plugging it in

Answer 21

argh you can show that it gets closer and closer to some number. if you want, you can call that number e

Answer 22

let's say \[e=\sum_{n=0}^\infty\frac{1}{n!}\] ;)

Answer 23

if i recall proof is not trivial.. you have to show it is increasing and bounded above. i think the bounded above part is easy though

Answer 24

very easy to show convergence with my def

Answer 25

how about
\[\int_0^e\frac{dt}{t}=1\] as a definition?

Answer 26

that one is nice too

Answer 27

what gets my goat is calc problems like this where you are supposed to take the log, use l'hopital, and then exponentiate and say "oh look, i got e!" going around in circles