Well, if you teach $\displaystyle \infty=+\infty$, 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 $\displaystyle \infty$ and $\displaystyle +\infty$ .
Fernando Revilla
I am indeed say exactly that.
The symbol $\displaystyle \displaystyle \lim _{x \to \infty } $ implies that if $\displaystyle N\in \mathbb{Z}^+$ then $\displaystyle x\ge N$.
Surely, that implies that $\displaystyle x>0$.
Out of curiosity, just what calculus text book did you use?
Right.
Fernando Revilla
I am not surprised because Bruce Edwards is at the University of Florida.
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 $\displaystyle \displaystyle \lim _{x \to \infty } $ means that x increases without bound.
@FernandoRevilla, I did not know that there were any scientific facts
If you take $\displaystyle \infty=+\infty$ then, nothig to say, I insist that you are completely right, but you are losing information about different kind of limits.
The reason is that $\displaystyle \mathbb {R}\cup \{\infty\}$ is homeomorphic to $\displaystyle S^1$ (the unit circumference) and $\displaystyle (0,1)$ acts as $\displaystyle \infty$. But $\displaystyle \mathbb{R}\cup\{+\infty,-\infty\}$ is homeomorphic to any close interval $\displaystyle [-a,a]\;(a>0)$ . Then $\displaystyle -a$ acts as $\displaystyle -\infty$ and $\displaystyle a$ acts as $\displaystyle +\infty$ .
Fernando Revilla