You have also

for a real variable a at your disposal.

Q: Is there also a DirectedInfinity[0]?

Hi, are you using Mathematica or Wolfram|Alpha?

It's possible to enter this symbol into Mathematica:[url=] reference.wolfram.com/mathematica/ref/PlusMinus.html
You can enter it into Mathematica by typing:
It is a symbol without any built-in meaning. You can define a meaning for it, but without it, Mathematica will not know what to do with it:
http://reference.wolfram.com/mathematica/tutorial/OperatorsWithoutBuiltInMeanings.html
I would evaluate the limit with both positive and negative infinity separately instead of defining the operator.

If you are looking to use Wolfram|Alpha, this has been asked once before here.
In short, you can't use Plus-Minus with Wolfram|Alpha. Enter each query individually:

lim (x^2+1)/(x+1) as x->infinity

lim (x^2+1)/(x+1) as x->-infinity