What we don't want to do, is list all possible definitions of this kind of limit (I count 15 of them). The key is understanding what are the ingedients that go into different kinds of limit, and how to "mix 'n' match" these ingredients to make any limit definition you need.
The first section here looks at limits as x approaches infinity, and the second section deals with left and right limits. There isn't very much on limits where the function goes to plus or minus infinity, but there is an example in Example 2.
This page emphasizes asking questions, and you should make an effort to try to write down the correct definitions yourself, before clicking to see the answers. There are a few extra questions at the end of the page, to test your understanding.
Note that the condition "D is not bounded above" is equivalent to saying "there is a sequence in D which converges to positive infinity". Thus this condition serves the same role in this definition as requiring that a be a limit point of D did in the original definition of limit.
Question: What do you think is the definition of the limit as x goes to minus infinity of f(x)?
Question: What do you think is the definition of the left limit of a function?
Question: What is the definition of