# 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.

• Formally those expressions don't make any sense, but think about what happens to $\frac1x$ if $x$ keeps getting smaller of bigger. Aug 16, 2011 at 18:45
• Aug 16, 2011 at 19:20
• Ok, I'll bite. Why is one equals sign squiggly and the other normal? Aug 16, 2011 at 21:00

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.

• Thanks for the reply. You have been very helpful. Sep 6, 2011 at 23:29
– Andy
Sep 7, 2011 at 4:19

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

These ideas are not really mathematical, but I believe the ideas are the following :

I think that idea behind $\frac{1}{\infty}$ is that $0 = \lim_{x \rightarrow \infty}\frac{1}{x} = 0$.

However, this reasoning would not work for the other, since $\lim_{x \rightarrow 0} \frac{1}{x}$ does not exists. However, the limit from the right is $\infty$. I think this depend on the nature of the question. For example if the value only take on positive real numbers.

• You could define a 'two-sided infinity' for which it will work out just fine. Aug 16, 2011 at 20:11

As already mentioned that's just shorthand for limits. Anyways suppose we have $1$ pizza and we want to divide this pizza equally into $n$ friends. Then each person gets a proportion: $\frac{1}{n}$ of the pizza.

Obviously the more friends you have the less each person is going to eat. And as your number of friends approaches $\infty$ then you get closer to eating almost nothing (proportion $=0$), hence:

$$\lim_{n \to \infty} \frac{1}{n}=0$$

Now for the other limit suppose we have $1$ string of length $1m$ and we divide it into $n$ equal parts. Let $x$ represent the length of each string, then in meters:

$$x=\frac{1}{n}$$

As we let $n \to \infty$ in other words if we divide the string into more and more parts, then the length $x$ approaches $0$.

This is equivalent to saying as we let the length $x$ approach $0$ from the positive numbers (right of zero) then we are dividing the string into more and more parts ($n \to \infty$). And hence:

$$\lim_{n \to 0^+} \frac{1}{n}=\infty$$