lim=infinity?

Question

I can't believe I'm asking this, because I should know this answer but I'm now doubting myself.

If a function goes to infinity as x approaches some real number, would we say the limit as x approaches that number is infinity or would we say that it does not exist?

Doesn't the limit always have to be a real number, but infinity isn't one?

So for example:
the limit as x approaches 0 from the right of 1/x is infinity or not defined?

Answer 1

No, in that case, the limit doesn't not exist. The symbol ∞ is not a number. We use the notation
lim x-->0+ 1/x = ∞.
This is only a smbolical way to express that we can make 1/x as large as we can.
We can read it though that
"the limit of f(x), as x approaches a. is infinity".

Maybe you can ask: if it doesn't exist, then why you say is infinity? To exist, it must be a real number.

p.s. my feeling is that ∞ can be regarded as a number but in an abstract sense. You add ∞ to the real line and say that it is the biggest element. I suppose that the manual tried to avoid this sophistication.

I guess what is important here is to understand the concept of limit. I guess concepts are not so strict and the manual is not a bible. But this is just my opinion.

Answer 2

we would say it is +infinity