Concept behind the limit to infinity? I can across transfinite numbers and came up with a thought.
What if$$\lim_{x\to\infty}f(x)=f(T)$$where $T$ was a transfinite number?
Generally, in calculus, I have noted that it is two different things, $\lim_{x\to a}f(x)\ne f(a)$.  Which makes me wonder about the situation $a=\infty$.  Which means it must mean something when we compare $\lim_{x\to\infty}f(x)$ and $f(\infty)$.  The two being different, I guessed the above statement about limits to infinite and transfinite numbers.
Because by definition:$$R<T<\infty$$where $R$ is the real numbers.
By definition:$$\lim_{x\to\infty_1^-}x<\infty_1$$Where the subscript indicates the two infinities are equal.  In general, the limit from the left side is close, but always less than the actual value.  At least, that is my understanding.
But at the same time, a limit to infinite is a limit by which $x$ exceeds real numbers.
So in a sense, $\lim_{x\to\infty}f(x)=f(T)$.
My question is whether or not my "postulation" is correct.
I have also noted that sometimes, on very rare occasions, the following is true:$$\lim_{x\to\infty}f(x)\ne f(\infty)$$
I will leave this up to you guys.

Adding larger points at infinity does not lead to anything useful beyond the effect of adding $\pm \infty$ to the system.  There is no known additional behavior of functions for large and growing finite real $x$ (which is what we want to analyze) that is not captured by adding $\infty$ or $\pm \infty$ to the domain, yet can be captured with additional transfinite infinite points.
Because the ordinals are discrete, there would not be any meaningful extension of $f(x)$ to values beyond $\omega$ (or $\infty$, or whatever the notation would be for the smallest infinite point added to the reals).  You could extend $f$ to be constant and equal to $f(\omega)$, or equal to $0$, or define $f(\omega + n)$ to equal $f(n)$ for all positive integers $n$, but none of these extensions would be useful for analysis of $f(x)$.
