Limits & Continuity

What this unit covers

The topics below follow the published Calculus BC course framework for Unit 1. This unit is worth 4–7% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• The Meaning of a Limit13 min · 3 objectivesInterpret a limit as the value a function approaches, independent of the value at the point · Distinguish two-sided limits from one-sided limits and state when a limit fails to exist · Evaluate limits from a graph, a table, and by direct substitution
• Continuity & Discontinuities13 min · 3 objectivesState the three-part definition of continuity at a point · Classify discontinuities as removable, jump, or infinite · Determine parameter values that make a piecewise function continuous
• Infinite Limits & Asymptotes14 min · 3 objectivesConnect infinite limits to vertical asymptotes · Evaluate limits at infinity by comparing degrees of numerator and denominator · Identify horizontal and vertical asymptotes of rational functions
• The Intermediate Value Theorem12 min · 3 objectivesState the hypotheses and conclusion of the Intermediate Value Theorem · Use the IVT to guarantee the existence of a root or a target value · Recognize why continuity on a closed interval is essential

Formulas in Unit 1

Every term in Unit 1

All 10 terms we publish for Limits & Continuity, with definitions. Reading them through is the fastest way to find the ones you cannot define — then drill those in cram mode until you can produce them without the prompt.

L'Hôpital's Rule conditions

Applies only to 0/0 and ∞/∞. Other indeterminate forms must first be algebraically rewritten into one of these.

Indeterminate forms requiring rewriting

0·∞ becomes a quotient; ∞ − ∞ needs a common denominator; 1^∞, 0⁰ and ∞⁰ are handled by taking logarithms first.

Repeated application of L'Hôpital

Legitimate as long as the form remains indeterminate after each step. Verify the form again before differentiating again.

Comparing growth rates

For large x, exponentials beat polynomials, which beat logarithms. This settles many limits at infinity without differentiating.

Three conditions for continuity at a

f(a) is defined, the limit exists, and the two are equal. All three belong in a justification.

Intermediate Value Theorem

A function continuous on [a, b] takes every value between f(a) and f(b) somewhere inside. Continuity on the closed interval must be stated.

Removable vs jump discontinuity

Removable means the limit exists but does not match f(a); a jump means the one-sided limits differ.

Limits producing horizontal asymptotes

Compare degrees for rational functions: denominator larger gives 0, equal gives the leading-coefficient ratio.

Squeeze theorem

If g ≤ f ≤ h near a and g and h share the limit L there, f has limit L too.

Special trig limits

lim(x→0) sin x/x = 1 and lim(x→0)(1 − cos x)/x = 0, which underpin the sine and cosine derivatives.

What examiners penalize here

• On the AP exam, always try substitution first and read the form. Continuous-looking value → done. 0/0 → simplify (factor/rationalize/special limit). Nonzero/0 → infinite limit (asymptote). Naming the form tells you the technique.
• When justifying continuity on the AP exam, cite all three conditions by name. Writing "f(a) is defined, the limit exists, and they are equal" earns the justification point that a bare numerical answer does not.
• For limits at infinity of rational functions, memorize the three-case degree rule — it turns most such problems into a one-line answer without any algebra. Reserve the divide-by-highest-power method for showing work when the exam demands justification.
• The IVT is an *existence* theorem — it never locates or counts solutions. If a prompt asks "how many" or "find the value," the IVT alone is insufficient; you would need additional tools like a sign analysis or the Mean Value Theorem.

Practice Calculus BC

Our practice bank is drawn from across the whole course rather than filtered to one unit, which is closer to how the exam asks anyway — it will not tell you which unit a question is testing.

Questions about this unit

All 10 units of AP Calculus BC

1. Unit 1 · Limits & Continuity
2. Unit 2 · Differentiation: Definition & Fundamental Properties
3. Unit 3 · Differentiation: Composite, Implicit & Inverse Functions
4. Unit 4 · Contextual Applications of Differentiation
5. Unit 5 · Analytical Applications of Differentiation
6. Unit 6 · Integration & Accumulation of Change
7. Unit 7 · Differential Equations
8. Unit 8 · Applications of Integration
9. Unit 9 · Parametric, Polar & Vector-Valued Functions
10. Unit 10 · Infinite Sequences & Series