Short on time? This page is the recap. For explanations, the graphs and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: lim (x → c) f(x) means “the limit as x approaches c of f(x)”.
Recap
- Definition in words. lim (x → c) f(x) = R if f(x) can be made as close to R as you like by taking x close enough to c, but not equal to c.
- The limit is one real number. If the outputs do not settle on one real number, the limit does not exist.
- Inputs on both sides of c count. x = c itself never counts.
- So the limit ignores f(c). f(c) can equal the limit, differ from it, or be undefined.
- One limit can be shown three ways: a graph (where the curve heads), a table (outputs closing in) and notation.
- A table suggests a limit; it does not prove it.
Key relationships
| Notation | Read it as | Meaning |
|---|---|---|
| lim (x → c) f(x) = R | “the limit as x approaches c of f of x is R” | Outputs near c, not at c, approach R |
| f(x) → R as x → c | “f of x tends to R as x tends to c” | Same statement, arrow form |
| f(c) = R | “f of c is R” | One output only; says nothing about the limit |
| lim (t → c) f(t) = R | (any letter) | Same as with x; the letter under lim must match the input |
| x → c⁻, x → c⁺, = ∞ | left, right, unbounded | Coming in Topic 1.3 |
Assumptions behind the definition
- f must be defined at points near c, on both sides (it need not be defined at c).
- “As close as you like” means for every closeness, not just one. The formal epsilon-delta version of this is not assessed.
Mistakes to avoid
- Writing “lim f(x) = R” with no “x → c” under lim.
- Dropping “lim” so the statement becomes f(x) = R.
- Swapping c and R: c is an input, R is an output.
- Using f(c) as the limit without a reason.
- Saying “f(c) is undefined, so there is no limit.” A hole does not stop a limit.
- Leaving out units and context when asked to interpret a limit.
Quick self-check
- Write in notation: “as x approaches 4, p(x) approaches 9”. (lim (x → 4) p(x) = 9)
- p(4) = 0 and lim (x → 4) p(x) = 9. Is this possible? (Yes. The limit ignores p(4). The graph has a hole at (4, 9) and a separate point at (4, 0).)
- W(d) is water depth in cm, d days into a drought. Interpret lim (d → 20) W(d) = 15. (As the time gets closer to day 20, the depth gets as close as we like to 15 cm.)
- A table shows f(2.999) = 6.0004 and f(3.001) = 5.9996. Does this prove lim (x → 3) f(x) = 6? (No. It suggests 6; a table shows only a few inputs.)
Next: practice questions.