What does it mean when a limit gives 0/0?
Updated
The short answer
Getting 0/0 when you substitute into a limit doesn't tell you the limit. It means you need to simplify first, often by factoring and canceling: (x² − 9)/(x − 3) equals x + 3 whenever x ≠ 3, so the limit as x → 3 is 6.
What it looks like
Here is the limit of (x² − 9)/(x − 3) as x → 3, worked two ways.
| Step | Wrong | Right |
|---|---|---|
| Substitute x = 3 | 0/0 | 0/0 |
| Next line | The limit is 0 | (x − 3)(x + 3)/(x − 3) = x + 3, for x ≠ 3 |
| Answer | 0 | 3 + 3 = 6 |
Another wrong ending is to write that the limit doesn't exist. You can check the right answer with numbers close to 3. At x = 2.9 the fraction is 5.9, and at x = 3.1 it's 6.1, so the values close in on 6 from both sides. The wrong column stops too early, at the line right after the 0/0.
In alai, an iPad app that reads your handwriting as you work, you can hold to talk and ask the tutor, Ally, why a step works.
Why it happens
Ordinary arithmetic offers three rules that seem to apply: 0 divided by anything is 0, anything divided by itself is 1, and you can't divide by 0. With 0/0 they point in different directions, and none of them settles it. Paul Dawkins's calculus notes call 0/0 an indeterminate form for this reason: the only way to know the limit is to work it out.
A limit doesn't ask what happens at x = 3. It asks what the fraction gets close to as x gets close to 3. The top and the bottom both shrink toward 0, and the limit depends on how they compare as they shrink. Here the top is (x − 3)(x + 3), so near x = 3 it's about 6 times the bottom.
Canceling the x − 3 is allowed because a limit only looks at values of x near 3, never at 3 itself, so x − 3 is never 0 there.
How to catch it yourself
When you get 0/0, take it as a sign to simplify before you plug in again. Point at the 0/0 and ask yourself:
What factor do the top and the bottom have in common?
The 0/0 itself is a clue. If a polynomial equals 0 at x = a, then x − a is one of its factors. If there's a square root, multiplying the top and the bottom by the conjugate often does the same job as factoring. It also helps to keep writing lim in front of each line until the step where you plug in, as Dawkins recommends.
Getting a nonzero number over 0, like 5/0, is a different case: then the limit isn't a finite number.
Try these
Find each limit.
- The limit of (x² − 4)/(x − 2) as x → 2
- The limit of (x² + 3x + 2)/(x + 1) as x → −1
- The limit of ((x + 5)² − 25)/x as x → 0
Answers
- 4, because the fraction equals x + 2 for x ≠ 2
- 1, because x² + 3x + 2 = (x + 1)(x + 2)
- 10, because (x + 5)² − 25 = x² + 10x, and (x² + 10x)/x = x + 10 for x ≠ 0
How alai helps
If you get 0/0 and aren't sure what to try next, hold to talk and ask, and Ally answers out loud with a hint. Ally also remembers what tripped you up in past sessions, so a later session can start from there.