Studying Mathematics Thread

[polymath257]

One way of understanding the epsilon-delta definition is in terms of tolerances.

Suppose you want to show that the limit of f(x) as x approaches a is L. Think of y=f(x) here.

We play a game: I give you a tolerance on y. That tolerance is called epsilon. This is how far from L I am willing to allow you to go.

You, in turn, have to give me a tolerance on x, which we call delta. This represents how far away from x=a you allow me to go.

Now, you win if *every* x value within your delta tolerance from 'a' produces y=f(x) value that is within my epsilon tolerance from L.

If, on the other hand, I can find an x that is *within* your tolerance, but where f(x) is *outside* of mine, then I win.

If the limit really is L, then you are guaranteed to win: you can always find a delta for any epsilon I pick.

I should say that understanding the epsilon-delta definition (and the corresponding epsilon-N definition for sequences) is a major step for those studying math. This one concept is what distinguishes a user of math from someone that actually understands it (well, up to a certain point). Almost everyone has trouble with it at first, so don't worry that all the 'for every, there exists, for every' stuff is tricky. Most people never have to go to that depth of quantifiers.