AP Calculus AB/BC Unit 1 Cheat Sheet: Limits & Continuity

The basics

What it covers: The limit — evaluating it graphically, numerically, and algebraically — plus continuity, discontinuities, asymptotes, and the Intermediate Value Theorem.

Exam weight: About 10–12% of the AB exam and 4–7% of the BC exam.

The big question: How do we describe what a function approaches, and when is a function continuous?

Mathematical practices: Implementing Mathematical Processes, Connecting Representations, Justification, and Communication & Notation.

Key topics at a glance

What Is a Limit

lim(x→c) f(x) = L: f(x) approaches L as x approaches c. It describes behavior near c, not at c — the limit can exist even where f(c) does not.

One-Sided Limits

Left: lim(x→c⁻); right: lim(x→c⁺). The two-sided limit exists only if both one-sided limits are equal.

Evaluating Limits

Direct substitution if continuous. For 0/0, use algebra: factor and cancel, or rationalize with the conjugate, then substitute.

Special Techniques

Squeeze Theorem: if g ≤ f ≤ h near c and lim g = lim h = L, then lim f = L. Key limit: lim(x→0) sin x / x = 1.

Continuity at a Point

All three: (1) f(c) defined, (2) lim(x→c) f(x) exists, (3) they are equal. Polynomials, exp, sin, cos are continuous everywhere.

Types of Discontinuity

Removable (hole; limit exists), jump (one-sided limits differ), infinite (unbounded → vertical asymptote).

Asymptotes from Limits

Vertical asymptote: lim = ±∞ (denominator → 0, numerator ≠ 0). Horizontal asymptote: lim(x→±∞) f(x) = L.

Intermediate Value Theorem

If f is continuous on [a, b] and k is between f(a) and f(b), then f(c) = k for some c in [a, b].

The key terms you must know

Limit — the value f(x) approaches as x → c; describes behavior near c.
One-sided limit — the limit from the left (c⁻) or right (c⁺).
Existence — a two-sided limit exists iff the two one-sided limits are equal.
Direct substitution — plug in x = c when f is continuous there.
Indeterminate form 0/0 — requires algebra (factor/rationalize) before substituting.
Squeeze Theorem — bound f between g and h with the same limit L.
sin x / x — lim(x→0) sin x / x = 1.
Continuity at a point — f(c) defined, limit exists, and they are equal.
Removable discontinuity — a hole where the limit exists.
Jump discontinuity — unequal one-sided limits.
Infinite discontinuity — unbounded behavior (vertical asymptote).
Vertical asymptote — where a one-sided limit is ±∞.
Horizontal asymptote — a finite limit at ±∞.
Intermediate Value Theorem — a continuous function attains every value between f(a) and f(b).

Key themes to remember

Limits are about approach, not arrival. What matters is behavior near c, not the value at c.
Try substitution first. If you get 0/0, do algebra (factor or rationalize) and try again.
Continuity has exactly three requirements. Check all three at the point in question.
Limits explain asymptotes. Infinite limits give vertical asymptotes; limits at infinity give horizontal ones.
The IVT needs continuity on a closed interval. State the hypotheses before you use it.

Common exam traps

A limit can exist where f is undefined. Don't assume the limit equals f(c) — check continuity separately.
0/0 is not the answer. It's an indeterminate form signaling you must simplify first.
Two-sided limits require agreement. If the left and right limits differ, the limit does not exist.
An infinite limit means the limit does not exist (as a finite number), even though we write it as ±∞.
Not every zero of the denominator is a vertical asymptote. If the factor cancels, it's a removable hole instead.
The IVT guarantees existence, not a specific location or the number of solutions.
Removable vs. jump. A removable discontinuity has an existing limit; a jump has unequal one-sided limits.