Limits & Continuity

What this unit covers

The topics below follow the published Calculus AB course framework for Unit 1. This unit is worth 10–12% of the exam, so budget your time against that rather than against how long the unit takes to teach.

Lessons in this unit

• One-Sided & Two-Sided Limits13 min · 3 objectivesInterpret a limit as the value a function approaches, independent of the value at the point · Evaluate left-hand and right-hand limits from graphs and piecewise rules · Determine when a two-sided limit exists by comparing the one-sided limits
• Algebraic Limits & the Squeeze Theorem14 min · 3 objectivesEvaluate limits by direct substitution and recognize the 0/0 indeterminate form · Resolve 0/0 limits by factoring, rationalizing, or simplifying · Apply the Squeeze Theorem to trap a limit between two known bounds
• Infinite Limits & Asymptotes14 min · 3 objectivesIdentify vertical asymptotes from limits that grow without bound · Evaluate limits at infinity by comparing degrees of numerator and denominator · Connect end behavior to horizontal asymptotes
• Continuity & the Intermediate Value Theorem13 min · 3 objectivesState the three-part definition of continuity at a point · Classify discontinuities as removable, jump, or infinite · Apply the Intermediate Value Theorem to guarantee a solution on an interval

Formulas in Unit 1

Every term in Unit 1

All 19 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.

Intermediate Value Theorem

If f is CONTINUOUS on [a, b] and k lies between f(a) and f(b), some c in (a, b) has f(c) = k. Continuity on the closed interval must be stated.

Limit definition informally

lim(x→a) f(x) = L means f(x) can be made arbitrarily close to L by taking x close enough to a. The value f(a) is irrelevant.

One-sided limits

The two-sided limit exists only when the left and right limits exist and are equal. A jump discontinuity is exactly where they differ.

Indeterminate form 0/0

Not an answer. Factor and cancel, rationalize, or apply L'Hôpital's Rule to resolve it.

Limits at infinity of rational functions

Compare degrees: denominator larger gives 0, equal gives the ratio of leading coefficients, numerator larger gives ±∞.

Horizontal asymptote

y = L when lim(x→±∞) f(x) = L. A graph may cross a horizontal asymptote, unlike a vertical one.

Vertical asymptote

Occurs where the limit is ±∞, typically where a denominator is zero and the numerator is not.

Removable discontinuity

A hole: the limit exists but does not equal f(a), or f(a) is undefined. Redefining one point repairs it.

Three conditions for continuity at a

f(a) is defined, the limit as x→a exists, and the two are equal. All three must be stated in a justification.

Squeeze theorem

If g(x) ≤ f(x) ≤ h(x) near a and g and h share a limit L at a, then f has limit L too. Used for x²sin(1/x) at zero.

Special trig limits

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

Continuity of piecewise functions

Set the two pieces equal at the boundary and solve for the parameter. Differentiability additionally requires the derivatives to match there.

Evaluating a limit algebraically

Substitute first. Only if that gives an indeterminate form do you factor, rationalize, or use a conjugate.

Limit of a piecewise function at a boundary

Evaluate the left and right limits from their own pieces. They must agree for the two-sided limit to exist.

Infinite limit vs limit at infinity

An infinite limit describes a vertical asymptote; a limit at infinity describes end behavior. The phrases are not interchangeable.

Limits involving absolute value

Rewrite |x − a| as a piecewise function and take one-sided limits, since the expression changes sign at a.

Showing a limit does not exist

Demonstrate that the one-sided limits differ, or that the function oscillates without settling. "It equals infinity" is a description of how it fails to exist.

Continuity on an interval

Continuous at every interior point, and one-sided continuous at the endpoints. Required before invoking IVT or EVT.

Extreme Value Theorem

A function continuous on a CLOSED interval attains an absolute maximum and minimum on it. The closed interval is the hypothesis that matters.

What examiners penalize here

• For piecewise functions, always compute the two one-sided limits separately using the correct branch, then compare. Writing "lim⁻ = lim⁺ = L, therefore the limit exists" is exactly the justification AP readers want to see.
• Reach for the Squeeze Theorem whenever you see a bounded factor (like sin or cos of something) multiplied by a factor going to 0. Bound the trig part between −1 and 1, multiply through, and let the outer limits pinch the answer.
• A horizontal asymptote describes end behavior *only* — a graph is allowed to cross its horizontal asymptote at finite x-values. Do not claim a curve "can never touch" y = L; that restriction applies to vertical asymptotes, not horizontal ones.
• A full-credit IVT argument has three moving parts: state that f is continuous on [a, b], compute both endpoint values to show the target lies between them, then conclude "by the IVT" a c exists. Skipping the continuity sentence loses the point even when the arithmetic is right.

Practice Calculus AB

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 8 units of AP Calculus AB

1. Unit 1 · Limits & Continuity
2. Unit 2 · Differentiation: Definition & Rules
3. Unit 3 · Composite & Implicit Differentiation
4. Unit 4 · Contextual Applications
5. Unit 5 · Analytical Applications
6. Unit 6 · Integration & Accumulation
7. Unit 7 · Differential Equations
8. Unit 8 · Applications of Integration