Indeterminate Forms, and Why Zero Over Zero Is Not a Number
Chapter Thirty-Seven
Syllabus topic Course Objective 6, "Inculcate the habit of Mathematical Thinking through Indeterminate forms" (named by no module label)
Pages 82 to 83 of 303
In one line
Zero over zero is not a number and not an error: it is a signal that the limit has to be worked out another way.
Why this chapter is in the book
MU's sixth Course Objective for this paper reads: "Inculcate the habit of Mathematical Thinking through Indeterminate forms". No module label mentions indeterminate forms at all.
That is recorded in this book's own contract rather than glossed over, and the topic is taught for two reasons. The internal twenty marks includes five for quizzes and assignments, which are set against the Course Objectives. And the topic is not decoration in this paper: the moment you have to show that the defining integral of a Laplace transform converges, or evaluate an initial-value theorem, or handle a partial fraction with a repeated root, you are taking a limit that arrives as an indeterminate form.
What "indeterminate" means
A form is indeterminate when knowing only the limits of the pieces does not tell you the limit of the whole.
Take a fraction whose top tends to zero and whose bottom tends to zero. What does the fraction do? The honest answer is: anything at all, depending on which of them gets to zero faster. Here are three fractions, all of the form zero over zero, with three different answers.
lim(x/x, x -> 0) = 1
lim((2x)/x, x -> 0) = 2
lim((x^2)/x, x -> 0) = 0
lim(x/(x^2), x -> 0) = oo
Four fractions of the same form, and the answers are 1, 2, 0 and unbounded. So the form tells you nothing, which is exactly what indeterminate means.
Compare that with a form that is not indeterminate. If the top tends to 5 and the bottom tends to 0 from above, the fraction grows without limit, and there is nothing to work out. That form is determinate.
The seven indeterminate forms
| Form | An example that gives one answer | Another that gives a different one |
|---|---|---|
| zero over zero | sin x over x tends to 1 | x squared over x tends to 0 |
| infinity over infinity | x over x tends to 1 | x squared over x grows |
| zero times infinity | x times one over x tends to 1 | x squared times one over x tends to 0 |
| infinity minus infinity | (x + 1) minus x tends to 1 | (2x) minus x grows |
| one to the power infinity | (1 + 1/x) to the x tends to e | 1 to the x tends to 1 |
| zero to the power zero | x to the x tends to 1 | needs care |
| infinity to the power zero | x to the 1/x tends to 1 | needs care |
Those seven are the whole list. Everything else is determinate.
Indeterminate Forms, and Why Zero Over Zero Is Not a Number
The forms that look indeterminate and are not
Students waste time on these, so they are worth naming.
| Form | Value | Why |
|---|---|---|
| a non-zero number over zero | grows without limit | the bottom is shrinking, the top is not |
| zero over a non-zero number | 0 | nothing to fight about |
| zero to the power of a positive number | 0 | |
| a positive number over infinity | 0 | |
| infinity plus infinity | grows without limit | both pull the same way |
| infinity times infinity | grows without limit |
Note that zero over zero is indeterminate but a number over zero is not. The difference is whether there is a competition. If the top is heading somewhere non-zero and the bottom is heading to zero, the bottom wins and there is no contest.
What to do with an indeterminate form
Three tools, in the order to try them.
One: factor and cancel. Usually the quickest, and it needs no theory.
lim((x^2 - 4)/(x - 2), x -> 2) = 4
lim((x^3 - 1)/(x - 1), x -> 1) = 3
Both of those are zero over zero at the point, and both come out at once by factorising the top and cancelling the common factor.
Two: divide through by the fastest-growing term. The standard treatment of infinity over infinity.
lim((3x^2 + 2x)/(x^2 - 5), x -> oo) = 3
lim((x + 1)/(x^2 + 1), x -> oo) = 0
Three: L'Hopital's rule, which is the next chapter, and which handles everything the first two cannot.
A form the rest of this module needs
Here is a limit of the kind this paper actually asks for, and it is zero over zero.
lim((1 - cos(x))/x^2, x -> 0) = 1/2
lim(sin(x)/x, x -> 0) = 1
lim((e^x - 1)/x, x -> 0) = 1
Those three appear constantly. The first two are needed when a Laplace transform is computed from a series; the third is the statement that the derivative of the exponential at zero is one, which is why the exponential is the function the transform is built on.
And here is an infinity-over-infinity limit that will appear in the chapter on the existence of the transform.
lim(x/e^x, x -> oo) = 0
lim(x^3/e^x, x -> oo) = 0
Any power of x, divided by an exponential, tends to zero. That single fact is the reason the Laplace integral of t to the n converges, and the reason MU's condition on s exists.
Check yourself
lim((x^2 - 9)/(x - 3), x -> 3) = 6
lim((5x^2 + 1)/(2x^2 - 3), x -> oo) = 5/2
lim(tan(x)/x, x -> 0) = 1
lim((1 + 1/x)^x, x -> oo) = e
lim(log(x)/x, x -> oo) = 0
The fourth is the definition of e written as a limit, and it is the standard example of the form one to the power infinity.
The rest of this subject
These notes are cut from the University's printed syllabus. Open the syllabus itself for the same subject.