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Old 09-20-2002, 12:14 PM   #1
RambunctiousRogue
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Default A Calculus Question


With all the geniuses around here, I figured someone could help me do a extra credit proof for Calculus.

Proove:

Lim [x(sin(1/x))] = 1
x->


Now for restrictions:
1) Brute Force Cannot Be Used
2) Verbal Proofs May Not Be Used

Someone told me it could be done by some brutal manipulation with trig-identities. It's extra credit because most of the time in Calc, you assume 1/ = 0, but is obviously not the case. If you can come up with it and show it, what can I say. You'll have my deepest respect and thanks.

Philchy
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Old 09-20-2002, 12:16 PM   #2
RambunctiousRogue
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Default whoopsy


sorry, the is supposed to be infinity (positive infinity that is).

Philchy
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Old 09-20-2002, 08:56 PM   #3
Stoolfoot
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Lim [x(sin(1/x))] = 1 = 0*
x->
let's assume is infinity
what you want is to get it 0/0 or / soo....
for 0* turn f*g into f/[1/g] or in this case [sin(1/x)]/[1/x]
then if you take the limit as x->, you'll end up with 0/0
now using L'Hospital's Rule take the derivative of both f and g separately sooo...1/x = u du=[-1/x^2]dx...d/dx sin u = cos u * du...you end up with...cos u du / du

lim ([cos(1/x)][-1/x^2]dx)/[-1/x^2]dx
x->

obviously [-1/x^2]dx cancel out and you are left with

lim cos(1/x)
x->

u = 1/x
lim 1/x = 0
x->

and finally here we go...

lim cos u = cos 0 = 1
u->0

oh yea!!!!!
this is using Calculus B and i'm assuming your in Calculus A or something lower so u might not even know about derivatives yet
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Old 09-21-2002, 06:59 AM   #4
Dauragon CMikado
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Note: Don't speak using "u" as "you" and "cos" as "because" if you're trying to explain a calculus problem.
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Old 09-21-2002, 07:41 AM   #5
Dwarkarn
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Fvck time for another beer I think....
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Old 09-21-2002, 07:07 PM   #6
Serien
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Another way to do it is to expand the sin function in a Taylor's series.

sin(u)= u - u^3/3! + u^5/5! ...

so

sin(1/x)= 1/x -1/(3! x^3) ...

then

x sin(1/x) = 1 - 1/(3! x^2) ...

therefore

lim [x sin(1/x)] = 1
x->inf

The "correct" way to solve the problem really depends on whether you've learned l'Hopital's rule or Taylor's series in your class yet.
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Old 09-23-2002, 03:51 AM   #7
Struzor
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Aye . . .beer . . .brain hurts . . .




Struzor Shadownite 53 Gnome SK
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