Well, if you teach , nothing to say. You are right with that writing convention.
What you have written above is not consistent with any standard calculus text book in use today. I have taught calculus classes since 1964.
That is not my problem. My reasons are scientific not related to habits. Besides a lot of books clearly distinguish between and .
I know the book well. Have used it many times. But your memory faulty.
In section 5 of chapter 3, there is a detailed discussion of limits ‘at infinity’.
In fact they say that means that x increases without bound.
@FernandoRevilla, I did not know that there were any scientific facts
The reason is that is homeomorphic to (the unit circumference) and acts as . But is homeomorphic to any close interval . Then acts as and acts as .