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Letto
Load Universeinto Cannon. Aimat Brain. Fire.
Registered: 12/13/02
Posts: 2,321

Calculus
#2573949  04/18/04 02:56 PM (12 years, 9 months ago) 


Since Shroomery doesn't have a math forum, I think this would be the best forum for me to ask for homework help. (I'm not asking for you to give me the work straight up, just any tips, so please don't flame me)
The assigned problem is Prove using L'Hopital's Rule that lim (n>infinity) (1 + 1/n)^n = e.
I first tried combining the term to get:
lim [(n+1)/n^2]^n = lim (n+1)^n/n^(2n)
but after taking the derivative (as per L'Hopital's Rule), I couldn't figure out anything, and enough of the terms didn't cancel.
So I tried to get the equality another way (not using L'Hopital's Rule) by setting
ln y = ln lim (n>infinity) (1 + 1/n)^n
ln y = n * lim ln (1 + 1/n)  ln 1/n = 0, so it's just ln 1 = 0
ln y = n * 0 = 0, but this cannot exist.
Can somebody be kind enough to tell me where I'm going wrong or if there's a better method for me to use (preferably using L'Hopital's Rule)? Any help is definitely appreciated!

luvdemshrooms
Two inch dick..but it spins!?
Registered: 11/29/01
Posts: 33,965
Loc: Lost In Space

Re: Calculus [Re: Letto]
#2573962  04/18/04 03:00 PM (12 years, 9 months ago) 


Blue. The answer is blue.
 You cannot legislate the poor into prosperity by legislating the wealthy out of prosperity. What one person receives without working for another person must work for without receiving. The government cannot give to anybody anything that the government does not first take from somebody else. When half of the people get the idea that they do not have to work because the other half is going to take care of them and when the other half gets the idea that it does no good to work because somebody else is going to get what they work for that my dear friend is the beginning of the end of any nation. You cannot multiply wealth by dividing it. ~ Adrian Rogers

MorbidHamster
Total Head Fook
Registered: 10/21/02
Posts: 121
Loc: UnUnited Kingdom
Last seen: 11 years, 8 months

Re: Calculus [Re: Letto]
#2574225  04/18/04 04:10 PM (12 years, 9 months ago) 


I think my brain just melted

Phencyclidine
Molecule
Registered: 06/02/03
Posts: 2,915

Re: Calculus [Re: Letto]
#2581287  04/20/04 03:41 AM (12 years, 9 months ago) 


Found it. I'll use x instead of n. inf. = "infinity"
[lim x to inf.] (1 + 1/x)^x
We have to change this to the form 0 / 0 or inf. / inf.
let y = (1 + 1/x)^x
thus [lim x to inf.] ln y = [lim x to inf.] x ln (1 + 1/x)
divide by (1/x) / (1/x)
so [lim x to inf.] ln y = [lim x to inf.] < ln (1 + 1/x) > / ( 1 / x)
which is of the form 0 / 0
apply L'Hopital's theorem
[lim x to inf.] ln y = [lim x to inf.] < 1 / (1 + 1/x) * ( 1 / x^2 ) > / ( 1 / x^2)
the ( 1 / x^2) terms cancel out, leaving
[lim x to inf.] ln y = [lim x to inf.] 1 / (1 + 1/x)
[lim x to inf.] e^(ln y) = [lim x to inf.] e ^< 1 / (1 + 1/x) >
[lim x to inf.] y = e^1
since y = (1 + 1/x)^x
[lim x to inf.] (1 + 1/x)^x = e
 


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