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 → a) f(x) means “the limit as x approaches a of f(x)”; x → a⁻ is from the left and x → a⁺ is from the right.
Recap
- Start from two facts: lim (x → a) c = c and lim (x → a) x = a.
- If lim f(x) = L and lim g(x) = M both exist, you can combine them with the properties below.
- Polynomials: lim (x → a) p(x) = p(a). Rational functions: substitute, as long as the bottom is not 0 at a.
- The properties also work for one-sided limits.
- Two-sided test: lim (x → a) f(x) exists only when both one-sided limits exist and are equal.
- The value f(a) never decides the limit.
Key relationships
| Property | Result | Condition |
|---|---|---|
| Sum / difference | L ± M | Both limits exist |
| Constant multiple | kL | lim f exists |
| Product | LM | Both limits exist |
| Quotient | L/M | Both exist and M ≠ 0 |
| Power | Lⁿ | n a positive whole number |
| Root | ⁿ√L | L > 0 for an even root |
| Composite | f(L), where g(x) → L | f can be evaluated by substitution at L |
Assumptions behind the method
- Every property assumes the separate limits exist and are finite.
- If a separate limit does not exist, the property is silent: work out each one-sided limit of the whole expression instead.
- If the outer function of a composite jumps at L, find which side g(x) approaches L from, and use that one-sided limit of f.
Mistakes to avoid
- Using the quotient property when the bottom limit is 0. Nonzero/0: no finite limit. 0/0: undecided (Topic 1.6).
- Assuming “no limit for f” means “no limit for f + g”. Jumps can cancel.
- Reading the filled dot as the limit. The filled dot is f(a); the limit is where the curve heads.
- Writing f(lim g) without checking f at that point.
- Losing signs: 3 − (−4) = 7 and (−2)³ = −8.
- Using the wrong piece of a piecewise function on one side of a.
Quick self-check
- Find lim (x → −2) (x³ + x)/(x − 1). (10/3: the top tends to −10 and the bottom to −3.)
- lim (x → 1) f(x) = 2 and lim (x → 1) g(x) = −1. Find lim (x → 1) [(f(x))²g(x) − 3g(x)]. (−1: that is 4 × (−1) − 3 × (−1) = −4 + 3.)
- f(x) = x² + 1 for x < 1, f(1) = 5, and f(x) = 2x for x > 1. Does lim (x → 1) f(x) exist? (Yes, it is 2: both one-sided limits are 2. The value f(1) = 5 does not matter.)
Next: practice questions.