Understanding Limits: Real Numbers or Infinity?

[buckr02]

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?

 

[EnumaElish]

It is a question of whether you are in the reals or the extended reals. The latter includes infinity.

 

[DeadWolfe]

Generically people will say the limit equals infinity. Indeed infinity is not a real number, but it is useful to use the shorthand of saying the limit equals infinity, as this is more specific than just saying that the limit is undefined.