# Analytical definitions vs intuitive (or perhaps first year ) definitions

1. Sep 9, 2010

### stripes

Analytical definitions vs intuitive (or perhaps "first year") definitions

I just began my real analysis course in college and we were given an assignment; a bunch of mathematical terms for us to define. We are asked to define them using two textbooks, one, our first year calculus textbook, the other, our real analysis textbook. The prof noted that there will be distinct differences between the two. He said "first year calculus books tend to use 'implications' as primary definitions, rather than precise definitions."

My problem is, the analysis textbook is very expensive and my current financial situation has not allowed for me to obtain the text immediately. So I can't look up anything in the analysis text.

My question is that I am unsure of what to look for...here, I'll give you an example.

We are asked to define the limit of a function at infinity: Let f be a function defined on some interval (a, ∞). Then lim┬(x→∞) f(x) = L means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large. For negatives, let f be a function defined on some interval (–∞, a). Then lim┬(x→-∞) f(x)=L means that the values of f(x) can be made arbitrarily close to L by taking x sufficiently large negative.

This is my definition from the first year book (stewart) that I chose. Now, this doesn't seem very "implicative" as my prof noted; to the contrary, it seems rather precise. Is the aforementioned definition the one that I would call the "analysis" definition? Or is this the "first year" definition? If this is simply the "first year" definition, then I am blown away by how much more math I have to learn (then again, either way, I know how much more there is to learn).

The way I see it, the above definition I gave seems to be the "real analysis" definition, and a simpler, "first year" definition would be something like: the limit of a function at infinity is finding the value of the limit of a function as its input becomes infinitely large.

I hope my question is clear, thank you all in advance and I apologize if this should have been posted elsewhere (eg homework forum).

2. Sep 9, 2010

### Landau

Re: Analytical definitions vs intuitive (or perhaps "first year") definitions

I don't really know what your prof means by first-year texts using "'implications' rather than precise definitions", as if implications are vague in general.

Anyway, I think this Stewart definition qualifies as a "first year" definition. It is not very precise because what do "arbitrarily close" and "sufficiently large" mean exactly? If you look at a real analysis book you'll see how to make this precise: the epsilon-delta definition.

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