Proving f(x) <= g(x): Choosing Epsilon

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In summary, the question is asking if lim [x->a] f(x) <= lim[x->a] g(x). The answer is that if f(x) is approaching a limit as x approaches a, then lim [x->a] f(x) will also approach a limit as x approaches a. However, if g(x) is approaching a limit as x approaches a, then lim [x->a] g(x) will also approach a limit as x approaches a.
  • #1
Buri
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The question is:

Suppose f(x) <= g(x). Prove that lim [x->a] f(x) <= lim[x->a] g(x).

I've been able to prove it by contradiction. I let lim [x->a] f(x) = L and lim[x->a] g(x) = M and I suppose that L > M. I then went on to choose epsilon = (L - M)/2 and a contradiction easily follows. But my problem is how do you arrive/choose this choice of epsilon? I just immediately thought of using this epsilon, but if I hadn't I don't know how/why I would have arrived at this epsilon. Can someone please help me out with this?

Thanks
 
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  • #2
You chose this [itex]\varepsilon[/itex] because it worked for the problem. The mental processes leading to this choice are probably extremely difficult. Experience certainly plays a big part.
 
  • #3
Not the answer I was hoping for. There is no way I could let's say draw a diagram and decide which epsilon would work and which one wouldn't?

For example, when you prove that f(x) can't approach two different limits I drew a diagram:

L________M

And I can "see" that epsilon = |L-M|/2 would make it impossible for both |f(x) - L| < epsilon and |f(x) - M| < epsilon

So I was hoping I could do something similar to understand my choice of epsilon.

Okay I'm going to try to explain this and I'd appreciate if you could tell me if this all makes sense.

I have:

M__________L

Why is it that I can't choose epsilon = |l-m|? Is it because then the intervals |f(x) - L| < |l-m| and |g(x) - M| < |l-m| overlap and hence I can't really tell whether g(x) < or > f(x)? However, when I do choose |l-m|/2 then |f(x) - L| < |l-m|/2 and |g(x) - M| < |l-m|/2 don't overlap anymore and so I will be guaranteed that g(x) < f(x) and hence a contradiction will follow somehow...

Do you see what I mean? I appreciate the help.
 
  • #4
I think the explanation/visualization with the overlapping intervals is appropriate.
 
  • #5
Thanks a lot for you help :)
 

FAQ: Proving f(x) <= g(x): Choosing Epsilon

How do you choose the value of epsilon for proving f(x) <= g(x)?

The value of epsilon is typically chosen based on the specific functions f(x) and g(x) being compared. In general, a smaller value of epsilon will provide a more precise proof, but it may also be more difficult to work with mathematically. It is important to choose a value of epsilon that is small enough to accurately represent the relationship between f(x) and g(x), but not so small that it becomes impractical to use in the proof.

What is the role of epsilon in proving f(x) <= g(x)?

Epsilon is a small positive number used to represent the margin of error in the proof. It acts as a boundary between f(x) and g(x), and helps to show that f(x) is always less than or equal to g(x) within a certain range. In other words, if we can prove that f(x) is always less than g(x) within an epsilon distance, then we can say that f(x) <= g(x).

Can epsilon be a negative number in proving f(x) <= g(x)?

No, epsilon must be a positive number in order to accurately represent the relationship between f(x) and g(x). A negative epsilon would imply that f(x) is greater than g(x) within a certain range, which would contradict the statement we are trying to prove.

How do you know if the value of epsilon used in the proof is valid?

The validity of the chosen value of epsilon can be checked by plugging it into the proof and making sure that all the steps and conclusions are mathematically sound. It is also helpful to see if the proof still holds when epsilon is slightly increased or decreased. If the proof remains valid with a small change in epsilon, then it is likely a valid value to use.

Are there any alternative methods for proving f(x) <= g(x) without using epsilon?

Yes, there are other methods for proving f(x) <= g(x) such as using limits or derivatives. However, epsilon is a commonly used and intuitive approach that is often preferred by mathematicians. It allows for a clear and straightforward way to show the relationship between two functions and can be easily understood by those who are not familiar with advanced mathematical concepts.

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