(rev. 6/16/2000)

Indeterminate forms are mathematical expressions that are meaningless as they stand. Common indeterminate forms include expressions of the form 0/0, ¥/¥, 1^¥, 0 · ¥, ¥⁰, ¥ – ¥, or 0⁰. There is simply no way to know whether the answer is 0, 1, ¥, or some other value that cannot easily be predicted. In calculus, students learn that indeterminate forms can sometimes have answers like e or 1/e as well.

For example, if someone asks you what the ratio of sin x to x is, when x is very close to 0, you have to approach the question with some caution. After all, when x is approximately 0, both sin x (the numerator) and x (the denominator) are approximately 0. We cannot state the answer as 0/0, because that is a meaningless, indeterminate form.

In case you’re curious, the answer is 1 (in this case, that is, not in general). If you pick a value for x that is close to 0 (say, 0.001), you can use a calculator to see that the answer is believable, since sin(0.001)/(0.001) = 0.9999998333 to the limits of your TI-83 display. By picking values of x even closer to 0, you quickly start to see what’s happening.

For a harder example, what is the limit of the ratio of 2x³ to 15x² as x approaches 0? Once again, we have an indeterminate form of the type 0/0, but this time, the answer is 0.

L’Hôpital’s Rule, which all calculus students encounter, is useful for resolving many types of indeterminate forms. In its most basic statement, L’Hôpital’s Rule covers only limits of the form 0/0 or ¥/¥ . However, by using clever algebraic tricks, one can often deal with indeterminate limits of the form 1^¥, 0 · ¥, ¥⁰, ¥ – ¥, or 0⁰ as well.

However, note that the following are not indeterminate forms. In each case, the symbols 0, 1, and ¥ on the LHS should not be interpreted literally. Here, they mean an expression which approaches 0 or 1, or which increases without bound. Remember, these are not indeterminate forms:
a/0 = +¥ if a is a fixed positive value
a/0 = –¥ if a is a fixed negative value
0/a = 0 if a is any fixed nonzero value
b/¥ = 0 for any finite value of b
¥/b = +¥ if b is a fixed positive value
¥/b = –¥ if b is a fixed negative value
c^¥ = 0 for any fixed c Î [0, 1)
c^¥ = ¥ for any fixed c > 1
1^c = 1 for any finite value of c
d · ¥ = ¥ if d is a fixed positive value
d · ¥ = –¥ if d is a fixed negative value
0 · d = 0 if d is any finite value
g⁰ = 1 if g is any fixed nonzero value
0^g = 0 if g is any fixed positive value
¥ + ¥ = ¥ always
similarly, ¥^¥ = ¥ and ¥ · ¥ = ¥
¥ – h = ¥ if h is any finite value
h – ¥ = –¥ if h is any finite value