The notion of $\infty$ in calculus

Question

According to calculus - an intuitive and physical approach(2nd) my Morris Kline 177p,

$$\lim\limits_{r_1\to \infty} \frac {GM}{r_1}\tag{66}\label{66}$$ We should note that we do not substitute $\infty$ for $r_1$. As we have previously noted, $\infty$ is not a number.
As $r_1$ becomes larger and larger, the fraction in (66) must become smaller because the numerator is a constant. Moreover, because $r_1$ takes on values such as $10^6, 10^{12}, ...,$ the fraction becomes as small and smaller than any small quantity one may name. That is, the fraction must come closer and closer to $0$ in value as $r_1$ becomes infinite. Then $$\lim\limits_{r_1\to \infty} \frac {GM}{r_1} = 0$$

If $\infty$ is not a number then $\lim\limits_{r_1\to \infty} \frac {GM}{r_1} = 0$ should be written as $\lim\limits_{r_1\to \infty} \frac {GM}{r_1} = \lim\limits_{x\to 0}x$. Because $GM$ is divided by $\infty$ which is not a number so I think the result is also not a number. Is my thought right?

Answer 1

No. The limit of the gravitational potential, $GM/r$, is $0$ as the distance $r$ goes to infinity. In other words, gravity has no effect on an object infinitely far away from it. That is the physical thought process.

Mathematically speaking: Yes, $\lim\limits_{r_1\to \infty} \frac {GM}{r_1} = \lim\limits_{x\to 0}x$ because they are both equal to zero. However, when you say:

...$\lim\limits_{r_1\to \infty} \frac {GM}{r_1}$ should be written as $\lim\limits_{r_1\to \infty} \frac {GM}{r_1} = \lim\limits_{x\to 0}x$. Because $GM$ is divided by $∞$ which is not a number so I think the result is also not a number

this is not a very good way to think about it. In calculus, the concept of a limit is the value of an input when its variable approaches a certain number. Saying that $\lim\limits_{r_1\to \infty} \frac {GM}{r_1}$ is not a number would simply incorrect.

Answer 2

No. By definition, the statement $$\lim_{r_1 \to \infty} \frac{GM}{r_1} = 0$$ means nothing more or less than for every real number $\varepsilon > 0,$ there exists a positive integer $M$ sufficiently large such that for all indices $r_1 > M,$ we have that ${\left\lvert \frac{GM}{r_1} \right\rvert} < \varepsilon.$ Essentially, this gives a precise way to say that $\frac{GM}{r_1}$ is "eventually arbitrarily small and positive."

Put another way, pick your favorite positive real number that is very small, e.g., $\varepsilon = 10^{-9}.$ If it is known that $$\lim_{r_1 \to \infty} \frac{GM}{r_1} = 0,$$ then I can find a positive whole number $M$ sufficiently large (maybe $10^{10^{10}},$ maybe smaller) such that for all positive whole numbers $r_1 > M,$ we have that $0 < {\left\lvert \frac{GM}{r_1} \right\rvert} < 10^{-9} = \varepsilon.$