Calculus help neede

February 15, 2007 7:45 PM

Quick help needed for my rusty calculus. Regarding algebraic functions, are the terms "no limit" and "limit does not exist" synonymous?
posted by Neiltupper to Education (9 answers total)

Probably, but the first phrase doesn't have any specific meaning for me. Can you give some examples?
posted by sbutler at 8:01 PM on February 15, 2007

If a function has no limit, you could interpret that to mean that the function goes to positive or negative infinity. However, consider the limit of cos(x) as x goes to infinity. This function is bounded between -1 and 1, but the limit does not exist.
posted by homer2k1 at 8:07 PM on February 15, 2007

Homer2k1 got it right, but I'd also like to point out that it's sometimes the case that functions tending to infinity are not regarding as non-existant limits. That is, a function can either converge to a number, positive or negative infinity, or diverge by oscillation, as cos(x) does at infinity, or sin(1/x) does at 0.
posted by TypographicalError at 8:21 PM on February 15, 2007

Best answer: yikes. A function cannot "converge to infinity". My high school calculus teacher taught me that lim(x^2) as x-> infinity = infinity (for example). That's a convenient way to think about it. But my college profs reminded me that such a function increases without bound, or without limit, so you should say that the limit does not exist.

Strictly speaking a function either has a limit (which must be a finite number) or it does not. So I would say "no limit" and "limit does not exist" are equivalent statements. Whether a function oscillates or grows monotonically is not relevant to the existance of a limit. Look at Example 6 on this page that I found for an example.

A good way to parse "limf(x) x-> a = b" is as follows:
If I make 'x' close to 'a', then as I get closer to 'a', the value of f(x) will get closer to 'b', and I can make f(x) as close to 'b' as I want simply by making 'x' closer to 'a'.

If this statement is not true then the limit does not exist. Note that if you plop in 'infinity' for 'b' it doesn't make sense - you can never make f(x) as close to infinity as you want because if you could it wouldn't be infinity. Er, hope that helped.
posted by PercussivePaul at 9:24 PM on February 15, 2007

Oh and I should comment on the distinction between bound and limit as homer2k1 pointed out. When we casually say a function grows "without limit" or that a function "has no limit", or "has no limits", we usually mean that, speaking mathematically, a function has no upper bound (or lower bound if it's negative). We would not normally say, in casual speech, that cosine grows without limit. But if you are talking about limits in a mathematical sense, the term 'limit' has a precise definition like the parsed version above and bounds are not in the picture.
posted by PercussivePaul at 9:34 PM on February 15, 2007

I would use the two phrases interchangeably (agreeing with PercussivePaul).
posted by em at 10:02 PM on February 15, 2007

I agree with PercussivePaul. Whether a limit doesn't exist because it goes to infinity or because it wiggles too much, it still doesn't exist. There may be a verbal distinction for some people, but I know I learned to answer "limit does not exist" for either situation. After all, there is no value such that for any epsilon there is a delta such that etc, etc. Now, there are topologies where you want to include the point/s at infinity to make them compact. The two most commons ones are probably the extended real line and the Riemann sphere, both of which allow infinity to be a well-defined limit.
posted by Schismatic at 10:06 PM on February 15, 2007

I'm pretty sure PercussivePaul has it right, but it's been a while since I've had to do a delta-epsilon proof -- "the horror".
posted by chunking express at 12:47 PM on February 16, 2007

Response by poster: Thank you for the responses. It certainly seems to me that they are synonymous and your comments tend to confirm that.

My question was prompted by the fact that structure of mathematics values the simpler of equivalents and yet I've been seeing the same two phrases, in the same chapters of text boks for what seem to be parallel examples yet with enough of a difference to prompt this question.
posted by Neiltupper at 9:13 PM on February 16, 2007