Question

lim (1 - 1/x)^(-x) as x->+infinty

Answer 1

= 0

Answer 2

can you please explain to me how you got that

Answer 3

I can.

Answer 4

if you let x = infinity. Then what is 1/x really equal to?

Answer 5

It's wrong though.

Answer 6

My solution? Please correct me then sir.

Answer 7

I think this is a special identity representing e, but let me check.

Answer 8

If my Cal I recall is correct, if the exponent was positive, it is the well-recognized lim that goes to e. The fact that exponent is is negative, I am leaning to agree with you it goes to 0. How did you reach your conclusion.

Answer 9

Well, You have something that is always smaller than one being raised to big negative numbers. It will certainly not tend toward 0 because the exponentiation will grow faster than the -1/x will shrink. However I think it does converge on e rather than go on to infinity.

Answer 10

It's 1-1, how do you get -1? And it's to the -x, meaning the equation is actually 1/(1-1/x) which is certainly zero.

Answer 11

No. It is
\[(1-\frac{1}{x})^{-x}\]
I agree that the fraction will tend toward 0, but the whole expression tends toward \(1^\infty\)

Answer 12

Actually I contend that inside the bracket goes to 1 because \[(1 \div \infty)\] tends to zero.

Answer 13

Oh my bad didn't see 1-1/x, even then the resultant inside is still zero. 0^infinity = 0.

Answer 14

it's not \(1-\infty\) it is 1- 0

Answer 15

God I need to stop failing and go to bed lol. I'm sorry polpak you're right.

Answer 16

But that is leaves us with \(\frac{1}{1^\infty}\) which is undetermined form and needs special handling with l'hopital.

Answer 17

Have to take the log to bring down the x out of the exponent, then take the derivative, etc. It's kind of a bear. It looks like it might be e though.

Answer 18

We need Lokisan for this one. We need someone who has read everything concerning the number or the limit e.

Answer 19

Heh. Well I checked with Alpha, and it's definitely e. Glad I don't have to do this for my homework though.

Answer 20

http://www.wolframalpha.com/input/?i=lim+as+x+goes+to+infinity+%281+-+1%2Fx%29^%28-x%29

Answer 21

Good job.