Let us consider two functions $f\left(t\right) = \frac{1}{t^2}$ and $g\left(t\right) = t^2$. We want to find the value of these two functions as $t\rightarrow \infty$. For the function $f$, this value is well defined and we write, $f\left(t\rightarrow\infty\right) = 0$. But for $g\left(t\right)$, the value lies at infinity. Which of the followings is true:
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The value of $g\left(t\rightarrow\infty\right)$ is not well defined.
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The value of $g\left(t\rightarrow\infty\right)$ lies at infinity.
Best Answer
If we were working in the extended real numbers, one would typically continuously extend both of your formulae to set $f(+\infty) = 0$ and $g(+\infty) = +\infty$, rather than presuming that you meant the formula to apply only for real $x$.
Doing the same with the projective real numbers instead, you would do the same and have $f(\infty)=0$ and $g(\infty) = \infty$.
Regarding the notion of limit, while it is traditional to say in the case that
$$ \lim_{x \to +\infty} x^2 = +\infty$$
that the limit is undefined, in my opinion that is a rather incomplete treatment of the situation: if you're going to have a concept of a limit equaling $+\infty$, then you ought to do the thing properly and have $+\infty$ as an actual mathematical object and say that the limit is defined and equals $+\infty$.