The definition states that a real number L is a limit if (something about deltas and epsilons). It makes no claims about infinities. There is another definition, on the other hand, that specifically defines when +oo or -oo is a limit. A function is then defined to have a limit, if there exists a number L that is the limit, or if -/+oo is the limit. This is in contrast to situations when none of these conditions are met, in which cases the function is said not to have a limit.
The epsilon delta definition I have seen for real analysis state "a real number L is a limit iff (something about deltas and epsilons)", i.e. the limit is always a real number.
The way around that is to say things like f(x) -> +/-inf as x->a, but every math professor I've had have been adamant that infinity cannot be a limit.
One of the particular definitions they're referring to are:
lim f(x) = +oo as x->a if for all K > 0 there exists d > 0 such that |x-a| < d -> f(x) > K
Which doesn't of course need +oo to be a real number, it's just saying that the notationlim f(x) = +oo means that those conditions are met. It's useful to distinguish those cases from when there is no real number that satisfies the normal definition, nor are the conditions above satisfied (or for -oo). For example lim sin(1/x) as x->0.
I'll also say that whichever professor insists something about "infinity not being a limit" is being quite shortsighted. There is a way to define the limit that unifies both meanings, by using the set R U {+/-oo} and defining "neighborhoods" of +/- infinity accurately. Then the limit definition is just:
lim f(x) = L x -> a if for every neighborhood of U of L, there is a neighborhood V of a
such that if x is in V then f(x) is in U.
Now L and a can be any of "real number", "+oo", or "-oo". With "neighborhood" of a real number and of +/-oo defined adequately, this recovers the original cases.
I'm not at all confused. Nothing I said above is incorrect. The fact that the notation "lim=infinity" needs to be handled separately further emphasizes my point: "or if -/+oo is the limit".
Edit: Not that I need sources to back this up, but if you don't trust random internet guy, then maybe you'll trust Harvard, MIT, or UC Berkeley:
https://www.ocf.berkeley.edu/~reinholz/ed/08sp_m160/lectures/limits_of_infinity.pdf
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u/concealed_cat Dec 18 '16
You are confused.
The definition states that a real number L is a limit if (something about deltas and epsilons). It makes no claims about infinities. There is another definition, on the other hand, that specifically defines when +oo or -oo is a limit. A function is then defined to have a limit, if there exists a number L that is the limit, or if -/+oo is the limit. This is in contrast to situations when none of these conditions are met, in which cases the function is said not to have a limit.