Why is $\frac{1}{\infty } \approx 0 $ and $ \frac{1}{0} = {\infty}$?

First I have checked the search option but found nothing relevant to my problem and also level of math. I just started learning the language of mathematics, on my own and I have trouble understanding why $ \frac{1}{\infty } \approx 0 $ and why $\frac{1}{0} = {\infty}$?

I want to know how exactly do I get to these answers. It might sound strange but it's hard for me as a beginner to picture things like this on my own.

Solution 1:

First of all, as @Jonas remarks in his comment, you should understand that those expressions are just a compact way of saying more complicated things, i.e. they are just symbols.

In the first case, you could try to interpret the infinity symbol $\infty$ as something very very large. Now, if you consider these fractions: $\frac{1}{1}$, $\frac{1}{10}$, $\frac{1}{10000}$ etc, you see that as the number in the denominator gets bigger, the number you are representing becomes smaller and smaller. This is formally understood as a limit: if you consider the function $f(x) = \frac{1}{x}$ and let $x \rightarrow \infty$ (which means that you let $x$ grow as much as you want), you will keep approaching $0$; thus, we write $\lim_{x \rightarrow \infty} \frac{1}{x} = 0$.

Now, for the second "equality", fractions work the other way around, consider this "sequence": $\frac{1}{\frac{1}{2}}$, $\displaystyle \frac{1}{\frac{1}{1000}}$, $\displaystyle \frac{1}{\frac{1}{10000}}$. You can easily see that what we actually have is $2,1000,10000$. Thus the numbers we obtain are progressively larger and larger. We could use the same approach here and consider $f(x) = \frac{1}{x}$ and let $x \rightarrow 0$ (which means we let $x$ get as small as we want), but as @William tells you this limit does not exist. This is actually a small technicality which I am not going to explain now, if you want more details please leave a comment and I'll expand, because -as you say- you want to understand how this goes. One way to save the day is to consider the function $f(x) = \frac{1}{x^2}$ and let $x \rightarrow 0$; this function behaves almost like the first one, but this time you will see that the numbers get bigger a lot faster as $x$ approaches $0$; in this case the technicality I was talking to you about doesn't present and we can write $\lim_{x \rightarrow 0} \frac{1}{x^2} = \infty$.

There are very precise definitions for what we mean by writing that, but the intuition behind it is this, and again I'm not going into much details since you don't say what "kind" of beginner you are. Again, if you want more details please leave a comment.

Solution 2:

The setting in which your two expressions are taken to be true, without too many extra conditions, is that of Möbius transformations in the complex numbers with a point at $\infty,$ together called the Riemann sphere. Anyway, see LINK