Mathematics
Indeterminate Forms Explained Intuitively
The topic of indeterminate forms is one that passes over Calculus 1 students as they are trying to solve for limits and integrals. Some of the most common ones are ∞/∞ , 0/0, ∞-∞, etc.
Most of the time, if not all, the teachers will only say that these terms are “indeterminate” without really proving it or explaining why it is. Let’s start with an example proof:
Zero Divided by Zero
For the expression in figure 1, let’s say that a = 10, c = 11 and b, d = 0. Now we have this:
After some rearranging we have this:
This kind of defies where intuition would lead us, though. Intuitively, or naively, 0 divided by 0 should be 1 — since most things divided by themselves is 1.
There was no error in our algebra, though. 0 divided by 0 is 10/11 in this case. But we could have picked any value for a and c, couldn’t we have? So realistically, 0/0 could be any value.
This is why we call it an indeterminate form. Because 0/0 can be any value we want it to be. This is why, if we approach a problem where we evaluate the limit of an equation, if we get 0/0 then we say that the result is indeterminate and we need to use L’Hôpital’s rule.
A nice informal definition of an indeterminate term is just that we don’t really know it’s exact value. We can’t determine it. It’s not determinate.
It’s actually saying that 0/0 could mean that the limit actually equals 1 or 5 or 1000 or whatever. We don’t know because 0/0 doesn’t really tell us anything, as we proved above.
A Thought Experiment with ∞
To wrap our heads around why ∞-∞ or ∞/∞ is indeterminate (the result is ambiguous), let’s start with a thought experiment. It’s called the Hilbert’s paradox of the Grand Hotel.
Imagine we have a hotel with a countably infinite number of rooms and a countably infinite number of people in those rooms.
Let’s say that a new person wanted to rent out a room in this infinite hotel. Are we to say that there is no more room because there are infinite people residing in the infinite number of rooms? Well, no.
The number of rooms is infinite (as in, there is no limit) so let’s just add one more for the new person, ∞ + 1 = ∞. This statement may defy common algebraic sense but remember that ∞ is not a number. ∞ is just a concept for a really huge number. What it’s really saying is:
(A big number) + 1 = (A big number) = ∞
By this logic, we can actually also say that a scalar times ∞ is also ∞.
Let’s try to prove that ∞-∞ is indeterminate now. By intuition, ∞-∞ should equal 0 . It could definitely take on that value but we can also prove that it doesn’t have to necessarily equal 0. Let’s say we have the following relationship:
This equation is basically saying, let’s take out half of the rooms in the hotel. This should still equal to infinity though since even if you halve infinity, it’s still infinity.
Let’s see another example where we take out all of the guests in the hotel except for 2 of them.
From these two examples, we’ve seen that ∞-∞ can have numerous different values (∞, 2, etc.). This is what makes ∞-∞ indeterminate, it’s ambiguous as to what value we’re actually talking about.
∞ divided by ∞
Let’s do one last common indeterminate form → ∞/∞. This one is actually relatively easy to prove given that we proved ∞-∞ is indeterminate. Let’s say we have the following relation:
Now let’s do some algebra to find ∞/∞.
We can see that ∞/∞ takes on two distinct values. Like all the other previous indeterminate forms, ∞/∞ can take on any value so we cannot actually determine what value of ∞/∞ we’re talking about.
Why do we care about this?
Indeterminate forms and their applications can be a bit abstract — it’s not everyday that I’m worrying that my improper integral has an indeterminate form.
Improper integrals are extremely important though, they appear a lot in science and engineering. A really famous example of the use of improper integrals is the Laplace transform or the Fourier Transform — algorithms that shape the world as we see it today.
So since we know improper integrals are important and abundant, indeterminate forms and knowing how to deal with them (L’Hôpital’s rule) are also inherently important.
Final Thoughts
I hope this shed some light on a topic that doesn’t receive too much explanation within the classroom. I also hope this was helpful and easy to understand. Thank you!
