1. ## disproving a limit.

I know how to prove something is a limit. I was wondering how to prove something isn't the limit using epsilon-delta. For example prove lim(x^2) as x->5 is not 15.

I am thinking I have to find an epsilon such that no d exists such that
0<|x-5|< d -> |x^2-15|< e.

What is the procedure?

Thanks.

2. ## Re: disproving a limit.

Originally Posted by Duke
I know how to prove something is a limit. I was wondering how to prove something isn't the limit using epsilon-delta. For example prove lim(x^2) as x->5 is not 15.
You really must understand the limit notion. If 15 were the limit then if we pick any number close to 5 then the square of that number is close to 15.

If $0 and $|x-5| then $16\le x^2$.
That means that $|x^2-15|\ge 1$. So we can always pick a number ‘close to’ 5 the square of which it not ‘close to’ 15.

3. ## Re: disproving a limit.

I know perfectly well that the limit is not 15. I just picked a very simple and obvious example so please don't be patronising. Maybe I should have picked a harder example. Btw your inequalitys should be strict since if |x-5|<1 then x cannot equal 4.

4. ## Re: disproving a limit.

I didn't see any patronizing. Why not just carry on with the discussion?

The point of epsilon-delta is that "very close to x" implies "very close to f(x)". It is sufficient to say, as Plato has indicated, that "x is close to 5" does NOT imply "f(x) is close to 15" when f(x) = x^2.

One way to demonstrate that the limit is NOT 15, is to demonstrate that the limit IS 25.

5. ## Re: disproving a limit.

I was objecting to this line. 'You really must understand the limit notion.' It is maybe my fault for picking such an obvious untruth but what I was trying to ask about was methods for more adavnced (still elementary I'm sure) problems. How can I, for example, get x as a function of delta? Then show that whatever delta is chosen, there exist F(x) not epsilon-close to L.

6. ## Re: disproving a limit.

Have you solved your problem now? Or? ...
If not, can you give an example where you're stuck or what's not clear.

7. ## Re: disproving a limit.

How can I get x as a function of delta?

8. ## Re: disproving a limit.

When the limit exists, it is convenient to say x = f(delta). What does it mean when the limit does not exist? Can we say x = g(delta) that shows us it is not so? I'm quite confused at what it is you seek.

More thought on the subject. Proving that a limit exists requires the removal of ALL examples to the contrary. Proving a limit does not exist requires ONLY ONE example to the contrary. These are quite different animals. Why do you expect a similar process to exist? Or, why would you expect the same result and thought process to produce both outomes? Your pursuit seems irrational.

9. ## Re: disproving a limit.

Originally Posted by Duke
I was objecting to this line. 'You really must understand the limit notion.' It is maybe my fault for picking such an obvious untruth but what I was trying to ask about was methods for more adavnced (still elementary I'm sure) problems.

Hi Duke, I don't think Plato was against what you picked as your untruth.

I do however think what Plato said was in a very different tone to how you read it. A simple misunderstanding that seems to happen a lot with written text.

10. ## Re: disproving a limit.

Originally Posted by TKHunny
When the limit exists, it is convenient to say x = f(delta). What does it mean when the limit does not exist? Can we say x = g(delta) that shows us it is not so? I'm quite confused at what it is you seek.

More thought on the subject. Proving that a limit exists requires the removal of ALL examples to the contrary. Proving a limit does not exist requires ONLY ONE example to the contrary. These are quite different animals. Why do you expect a similar process to exist? Or, why would you expect the same result and thought process to produce both outomes? Your pursuit seems irrational.
delta still controls the distance between x and a so I would have thought it may lead to a useful method. Are there tricks that are often useful in proving limits don't exist or similar question.

11. ## Re: disproving a limit.

I mentioned one, already. Prove what it IS and all the "Isn't"s are done simultaneously.

Personally, I would resort to the definition of a limit. If it doesn't exist, it is likely to violate some premise of the definition.

12. ## Re: disproving a limit.

You could formally (in the sense of logic) negate the statement that the limit as x approaches y of f(x) is L.

The formal statement is "for every epsilon there exists a delta such that for every y with |x-y|<delta, |f(x)-L|<epsilon".

You negate a statement by flipping all the quantifiers (changing "for every" to "there exists" and vice versa) and flipping the inequality in the conclusion. So the negated statement is "there exists some epsilon so that for every delta there exists a y with |x-y|<delta and |f(x)-L|>=epsilon".

Now prove the negation is true, and you've proved the original statement false.

Edit: While it's true that showing the limit is 25 proves that it cannot be 15, that relies on specific facts about the real numbers (namely that R is a Hausdorff space and so limits are unique). When in doubt, it's always best to prove exactly what you are asked for, i.e. that the statement "the limit is 15" is false.