Limit Comparison Theorem: Proving the Relationship Between Two Functions

[cristina89]

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??

 

[phinds]

Does the limit of a function ever give you a value that is higher than the range of values of the function itself?

 

[HallsofIvy]

Try a "proof by contradiction". Suppose lim f(x)> lim g(x). Let \alpha= lim f(x)- lim g(x) and choose \epsilon= \alpha/2.

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.)

 

[phinds]

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.

 

[LCKurtz]

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 $\lim_{x\to\infty}\frac 1 x$.

 

[phinds]

Try $\lim_{x\to\infty}\frac 1 x$.

OK, thanks. I got it.