AP® Calculus BC · Unit 1 · Topic 1.2
Defining Limits and Using Limit Notation
A limit describes the value a function approaches as x gets arbitrarily close to a point, based only on nearby behavior, not on what happens exactly at that point.
A limit asks one question: as x sneaks up on some value a, from both sides, what height does f(x) settle near? Notice what that question does not ask. It never asks what f(a) actually is. That's why a function can have a perfectly good limit at a point where it is undefined, has a hole, or jumps to some other value entirely. The limit only cares about the neighborhood around a, never the single point itself.
The notation lim(x→a) f(x) = L packages this idea: it says that as x approaches a (but never equals a), f(x) approaches L. Take (x2-4)/(x-2) at x=2. Plug in 2 directly and you get 0/0, a warning sign, not an answer. But factor the numerator as (x-2)(x+2), and for every x not equal to 2 the expression simplifies to x+2. The hole at x=2 is real, but all around it, f(x) is behaving exactly like x+2, so the limit is 4, even though f(2) itself is undefined.
This separation between 'what the limit is' and 'what the function value is' is the whole point of the topic, and it sets up continuity, which is the special case where the two finally agree. It also sets up the derivative: a derivative is built from a limit of a difference quotient, (f(x)-f(a))/(x-a), which is itself 0/0 at x=a. Everything you practice here, chasing a trend without ever evaluating at the forbidden point, is exactly the skill Unit 2 needs.
Key terms
- limit
- The value a function approaches as the input gets arbitrarily close to a point, without needing to reach it.
- removable discontinuity (hole)
- A point where a function is undefined or disagrees with its limit, but the limit still exists.
- two-sided limit
- A limit that requires the function to approach the same value from both the left and the right of a point.
- indeterminate form
- An expression like 0/0 that gives no direct information and signals the need for algebraic simplification.
Watch it explained
- How it worksHow does limit notation describe what happens near a point without ever touching it?
- Worked exampleFind the limit of (x2 - 4)/(x - 2) at x = 2 even though there is a hole sitting there.
- Common mistakeWhy doesn't the limit at x = 3 have to equal the function's actual value at x = 3?
- Ties togetherHow does this two-sided limit turn into the difference quotient that defines the derivative in Unit 2?
Check yourself
End of topic.