Limits & Continuity

What this unit covers

The topics below follow the published Calculus BC course framework for Unit 1. This unit is worth 5–10% 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 22 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.

Growth hierarchy for limits

ln x ≪ any power of x ≪ any exponential aˣ (a > 1) ≪ n! for sequences. Settles most limits at infinity without a single application of L'Hôpital.

Why limits matter twice in BC

Every convergence test is a limit — the ratio test, the nth-term test, and the improper integrals behind the integral test. Weak limits cost points in Unit 10, not Unit 1.

0·∞ requires rewriting

Convert to a quotient before applying L'Hôpital, and choose which factor goes underneath: the one whose reciprocal is easier to differentiate.

1^∞ and 0⁰ forms

Take the natural log of the expression, find the limit of the logarithm, then exponentiate. The answer to the original is e raised to that limit.

∞ − ∞ form

Combine into a single fraction, or rationalize, until the expression is a quotient. Only then is L'Hôpital available.

Recheck the form after each L'Hôpital step

The rule may be applied repeatedly, but once the result is determinate you must stop and evaluate. Continuing produces a wrong answer from correct arithmetic.

One-sided limits and existence

lim(x→a) f exists if and only if both one-sided limits exist and are equal. A piecewise function whose branches disagree at the seam has no two-sided limit.

Oscillation as a failure mode

sin(1/x) has no limit at 0 because it crosses every value in [−1, 1] infinitely often. Bounded, yet no limit — a distinct failure from a jump or a blow-up.

Limit at infinity by degree

Numerator degree below denominator gives 0; equal degrees give the ratio of leading coefficients; numerator above gives no finite limit.

Continuity of a composition

If g is continuous at a and f is continuous at g(a), then f∘g is continuous at a — which is why compositions of polynomials and trig functions are continuous where defined.

Limit of a sequence versus of a function

A sequence limit takes n through the integers only, so lim aₙ can exist where the corresponding function limit does not. Useful when applying L'Hôpital to sequences.

Why a table cannot prove a limit

A table samples finitely many points. It supports the word "estimate" and never the word "justify"; for justification use algebra or a theorem.

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