Proof of theorem (Limit)

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Homework Statement


Prove that if f(x)<=g(x) then lim f(x) <= lim g(x).


Homework Equations


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The Attempt at a Solution



I've tried by definition of limit, but I didn't get anywhere with this... Can anyone help me??
 

Answers and Replies

  • #2
phinds
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Does the limit of a function ever give you a value that is higher than the range of values of the function itself?
 
  • #3
HallsofIvy
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Try a "proof by contradiction". Suppose lim f(x)> lim g(x). Let [itex]\alpha[/itex]= lim f(x)- lim g(x) and choose [itex]\epsilon= \alpha/2[/itex].

I am puzzled by phind's question. The answer is "yes, it does" but I don't see how that helps here.

(Note, by the way, if the condition were "f(x)< g(x)" then it would NOT be true that "lim f(x)< lim g(x)". Phind's suggestion would be helpful in proving that.)
 
  • #4
phinds
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I am puzzled by phind's question. The answer is "yes, it does" but I don't see how that helps here.

QUOTE]

Hm ... I guess I'm missing something. I don't see how the limit of a function could possibly give you a value that is outside the range of the possible values of the function.
 
  • #5
LCKurtz
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Hm ... I guess I'm missing something. I don't see how the limit of a function could possibly give you a value that is outside the range of the possible values of the function.
Try ##\lim_{x\to\infty}\frac 1 x##.
 
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  • #6
phinds
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Try ##\lim_{x\to\infty}\frac 1 x##.
OK, thanks. I got it.
 

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