What is a limit in calculus?
Think about where a graph is heading, not only where it lands. Approaching from both sides is the key to a two-sided limit.
Try the missing-point example
Let f(x) = (x² − 4) / (x − 2). At x = 2, direct substitution gives 0/0, so the original expression is undefined there. For every x other than 2, factor the numerator:
(x² − 4) / (x − 2) = (x − 2)(x + 2) / (x − 2) = x + 2, for x ≠ 2.
The nearby values follow the line y = x + 2. The missing point is a hole, but the graph still heads toward y = 4 as x heads toward 2.
| x | 1.9 | 1.99 | 2.01 | 2.1 |
|---|---|---|---|---|
| f(x) | 3.9 | 3.99 | 4.01 | 4.1 |
limx→2 (x² − 4) / (x − 2) = 4.
How to read the answer
The limit is 4; the function value at x = 2 is undefined. Those statements are compatible. A common mistake is treating 0/0 as the limit. It signals that you need another step, such as factoring, simplifying, or reading the nearby graph.
For a two-sided limit, the values from the left and right must approach the same number. See what happens when they do not.
See the calculus move
Calculus Animated pairs cartoon lessons with simulations and practice. Start with two complete lessons for free. Email confirmation is required; no credit card is needed.
Try two lessons →