Forum for the GRE subject test in mathematics.

gaucho85
 Posts: 11
 Joined: Thu Sep 18, 2008 5:58 pm
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by gaucho85 » Thu Oct 16, 2008 3:46 pm
can anyone give an example of functions f and g such that
lim x> inf of f/g = 1 but lim x>inf of e^f/e^g is not 1?
That's what 29 is saying, right? Thanks.
Last edited by
gaucho85 on Thu Oct 16, 2008 4:08 pm, edited 1 time in total.

moo5003
 Posts: 36
 Joined: Mon Oct 06, 2008 7:33 pm
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by moo5003 » Thu Oct 16, 2008 4:01 pm
f = sin(x)
g = x
f/g tends to 1 (check with lhospitals rule)
Edit: Sorry I was thinking as x tended toward 0 Ignore everything
.
e^f oscillates between [1/e,e]
x^sin(x) oscillates as well but with increasing amplitude end up going from [0,Infinity]
thus e^f / g^f does not even converge as x tends to infinity
Last edited by
moo5003 on Thu Oct 16, 2008 5:41 pm, edited 1 time in total.

gaucho85
 Posts: 11
 Joined: Thu Sep 18, 2008 5:58 pm
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by gaucho85 » Thu Oct 16, 2008 4:11 pm
thanks for all your help, moo.
I'm still not quite understanding this. I had a typo in my original question, which I've fixed, maybe that caused some confusion.
the lim as x> inf of sin(x)/x goes to 0, not 1, right? (sin is bounded, x gets arbitrarily large).

Nameless
 Posts: 128
 Joined: Sun Aug 31, 2008 4:42 pm
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by Nameless » Thu Oct 16, 2008 4:13 pm
lim_(x>0)(sinx)/x=1 . This one is very famous, you can prove it using L'Hopistal rule

gaucho85
 Posts: 11
 Joined: Thu Sep 18, 2008 5:58 pm
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by gaucho85 » Thu Oct 16, 2008 4:28 pm
thanks everyone. however, this problem talks about the limit as x goes to infinity, not zero.
Is there any way one can just "tell" that the answer is C, or do you find a specific counterexample where C is not implied from the problem?

Nameless
 Posts: 128
 Joined: Sun Aug 31, 2008 4:42 pm
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by Nameless » Thu Oct 16, 2008 4:34 pm
Okie
let f(x)=x^2+x
g(x)=x^2
then f/g>1 when x>infinity
e^f/e^g=e^(fg)=e^x>infinity when x>infinity

aspirant
 Posts: 7
 Joined: Thu Nov 06, 2008 2:51 am
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by aspirant » Thu Nov 06, 2008 3:41 am
A followup to this question: why is (D) always correct?
lim(x>inf) (f/g) = 1 always implies lim(x>inf) ((f+g)/2g) = 1
Can someone explain the above please? Thanks...

Nameless
 Posts: 128
 Joined: Sun Aug 31, 2008 4:42 pm
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by Nameless » Thu Nov 06, 2008 10:00 am
lim(x>inf) (f/g) = 1
then
lim(x>inf) ((f+g)/2g) = lim(x>inf) (f/2g+g/2g)=1/2+1/2=1

aspirant
 Posts: 7
 Joined: Thu Nov 06, 2008 2:51 am
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by aspirant » Fri Nov 07, 2008 12:19 am
It's just algebra! Thanks Nameless! This test really requires a cool mind to see the problem from different angles  especially under exam stress.