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Limits at infinity, lim xF(x) = L then lim (f(x)=0 
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#1
Mar1412, 01:06 PM

P: 108

1. The problem statement, all variables and given/known data
show that if F:(a,∞) >R is such that lim xF(x) = L, x > ∞, where L is in R, then lim F(x) = 0, x > ∞. 2. Relevant equations 3. The attempt at a solution Let F:(a,∞) →R is such that lim xF(x) = L, x → infinity, where L is in R. Then there exists an α> 0 where given ε, there exist k(ε) for all x > k then ε > max{1 , ([L]+1)/x} Therefore [xF(x)  L] < 1 whenever x > α. Therefore [F(x)] < ([L]+1)/x. Thus [F(x)0] < ε Then lim F(x) =0 as x → ∞. This is what I have but it doesn't look right to me. 


#2
Mar1512, 06:16 PM

P: 108

Proof: let f:(a,∞)→ℝ such that lim xf(x)=L where L in ℝ. Since lim xf(x) = L, there exists α>0 where xf(x)L < 1 for all x > α. Therefore f(x)<(L+1)/x for x >α. Pick ε = m where there exist δneighborhood V_{δ}(c) of c and x is in A π V_{δ}(c), there exists m>0, m = L+1 then f(x) < M, for all X, therefore f(x)o<M=
thus the limit_{x→∞} f(x) =0. Does this proof make more sense? Am i still missing something? 


#3
Mar1512, 06:40 PM

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You can't simply pick the ε in the part where you prove that lim_{ x→∞ }f(x) = 0 . It may help for you to state, in ε  M language, what it means that lim_{ x→∞ }f(x) = 0 . 


#4
Mar1512, 06:47 PM

P: 297

Limits at infinity, lim xF(x) = L then lim (f(x)=0
This can be proved in one line.
Lt x>infinity xf(x) =L (where L is finite) So Lt x>infinity f(x) = L/x (How?) What do you see?? 


#5
Mar1512, 06:53 PM

P: 108




#6
Mar1512, 07:17 PM

P: 297

Correct :)



#7
Mar1512, 10:04 PM

P: 108




#8
Mar1612, 02:01 AM

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#9
Mar1612, 06:01 AM

P: 108




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