Understanding Epsilon Delta Definitions for Limits: Functions that Satisfy Them

[morbius27]

Homework Statement

Im trying to figure out what the difference is between the following two epsilon delta statements and the kinds of functions they satisfy:

For all real numbers x and for all delta>0, there exists epsilon>0 such that |x|<delta implies |f(x)|<epsilon

vs.

there exists delta>0 such that for all epsilon>0 and for all real numbers x, |x|<delta implies |f(x)|<epsilon

I'm just very confused about the whole epsilon delta thing. I looked online and found the definition of a limit and tried to understand what part epsilon and delta played in the definition, but things like the FOR ALLs and the apparent importance of order in the definition are confusing me as to what exactly they're trying to say.

 

[Matterwave]

Both statements practically mean that the function tends to 0 as x tends to 0 on both sides. I believe the first statement is the correct way of saying this limit.

 

[morbius27]

Both statements practically mean that the function tends to 0 as x tends to 0 on both sides. I believe the first statement is the correct way of saying this limit.

But wouldn't the change in order of the epsilon and delta plus the switch between there exists and for all change the meaning of the statements?

 

[holezch]

yes, there is a difference, they are both incorrect. The ordering will matter because it is the order you "pick" what you want. So, for example, if x came after delta, you aren't able to pick delta in terms of x.

it should be

for any epsilon > 0, there exists a delta >0 such that if 0 < |x - a| <delta, then |f(x ) - L | < epsilon.

for the placing of x and a, you could say

for any a in |R, for any epsilon > 0, there exists a delta >0, for any x in |R such that if 0 < |x - a| <delta, then |f(x ) - L | < epsilon.

epsilon is the "margin of error" for the values produced by a function. If x actually approaches L as x approaches a, then you should be able to find values as close to L as you want. so you should be able to find values close to L with closeness defined as "epsilon", if you can really come as close as you want, then you should be able to pick ANY epsilon you want and have it work. This is the meaning behind FOR ANY epsilon

 

[Tedjn]

Actually, I can't really make heads or tails out of either definition. Are you sure they are correct?

For the first, let's reason about what it actually says. First of all, the for all real numbers x part is redundant. This is because you then choose a delta and only consider |x| < delta. Clearly, then, you are not considering all real numbers x. Hence, already, we lose nothing by shortening the first statement into:

For all $\delta > 0$, there exists $\epsilon(\delta) > 0$ such that $|x| < \delta \implies |f(x)| < \epsilon$.
​

Here, I explicitly made it clear that we choose $\epsilon$ based on our initial choice of $\delta$. We choose our $\epsilon$ such that whenever |x| is within $\delta$ distance of 0, our function is bounded by $-\epsilon$ and $\epsilon$.

This is not the limit definition; indeed all it says is that our function f(x) is bounded inside the interval $(-\delta, \delta)$. For we can choose $\epsilon$ as large as we want; and as long as f(x) doesn't go off to infinity in a vertical asymptote, we're fine.

We can also examine the second statement, but upon a glance it also seems wrong. What most people question when they consider order of quantifiers is the difference between the definitions for continuity and uniform continuity. Is that what you are actually asking about?

 

[Matterwave]

yes, there is a difference, they are both incorrect. The ordering will matter because it is the order you "pick" what you want. So, for example, if x came after delta, you aren't able to pick delta in terms of x.

Although this applies to statement 2, I don't see how this applies to statement 1.

EDIT: I see my mistake. Disregard all of my statements in this thread.

 

[morbius27]

Well what I'm trying to ask is that given the statements in the initial post, what set of functions From R -->R satisfy them. The reason I ask about order is because i feel like the order in which the parts of the statements come affects the entire statement as a whole. The statements are correct as far as they were given to me to analyze--I'm just not quite sure what set of functions they are describing and what the difference between them is.