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# Limits and continuity

AP Calc: CHA (BI), CHA‑1 (EU), FUN (BI), FUN‑1 (EU), LIM (BI), LIM‑1 (EU), LIM‑2 (EU)

AP Calc: LIM (BI), LIM‑1 (EU), LIM‑1.A (LO), LIM‑1.A.1 (EK), LIM‑1.B (LO), LIM‑1.B.1 (EK)

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Limits introLimits intro

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Limits introGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑1 (EU), LIM‑1.C (LO), LIM‑1.C.1 (EK), LIM‑1.C.2 (EK), LIM‑1.C.3 (EK), LIM‑1.C.4 (EK)

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Estimating limit values from graphsUnbounded limitsEstimating limit values from graphsOne-sided limits from graphsOne-sided limits from graphs: asymptoteConnecting limits and graphical behavior

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Estimating limit values from graphsGet 3 of 4 questions to level up!

One-sided limits from graphsGet 3 of 4 questions to level up!

Connecting limits and graphical behaviorGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑1 (EU), LIM‑1.C (LO), LIM‑1.C.5 (EK)

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Approximating limits using tablesEstimating limits from tablesUsing tables to approximate limit valuesOne-sided limits from tables

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Creating tables for approximating limitsGet 3 of 4 questions to level up!

Estimating limits from tablesGet 3 of 4 questions to level up!

One-sided limits from tablesGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑1 (EU), LIM‑1.D (LO), LIM‑1.D.1 (EK), LIM‑1.D.2 (EK)

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Limit propertiesLimits of combined functionsLimits of combined functions: piecewise functionsLimits of composite functions

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Limits of combined functions: sums and differencesGet 3 of 4 questions to level up!

Limits of combined functions: products and quotientsGet 3 of 4 questions to level up!

Limits of composite functionsGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑1 (EU), LIM‑1.D (LO), LIM‑1.D.1 (EK), LIM‑1.D.2 (EK)

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Limits by direct substitutionUndefined limits by direct substitutionLimits of trigonometric functionsLimits of piecewise functionsLimits of piecewise functions: absolute value

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Limits by direct substitutionGet 3 of 4 questions to level up!

Direct substitution with limits that don't existGet 3 of 4 questions to level up!

Limits of trigonometric functionsGet 3 of 4 questions to level up!

Limits of piecewise functionsGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑1 (EU), LIM‑1.E (LO), LIM‑1.E.1 (EK)

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Limits by factoringLimits by rationalizingTrig limit using Pythagorean identityTrig limit using double angle identity

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Limits by factoringGet 3 of 4 questions to level up!

Limits using conjugatesGet 3 of 4 questions to level up!

Limits using trig identitiesGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑1 (EU), LIM‑1.E (LO), LIM‑1.E.1 (EK)

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Strategy in finding limitsStrategy in finding limits

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Conclusions from direct substitution (finding limits)Get 3 of 4 questions to level up!

Next steps after indeterminate form (finding limits)Get 3 of 4 questions to level up!

Strategy in finding limitsGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑1 (EU), LIM‑1.E (LO), LIM‑1.E.2 (EK)

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Squeeze theorem introLimit of sin(x)/x as x approaches 0Limit of (1-cos(x))/x as x approaches 0

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Squeeze theoremGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑2 (EU), LIM‑2.A (LO), LIM‑2.A.1 (EK)

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Types of discontinuities

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Classify discontinuitiesGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑2 (EU), LIM‑2.A (LO), LIM‑2.A.2 (EK)

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Continuity at a pointWorked example: Continuity at a point (graphical)Worked example: point where a function is continuousWorked example: point where a function isn't continuous

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Continuity at a point (graphical)Get 3 of 4 questions to level up!

Continuity at a point (algebraic)Get 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑2 (EU), LIM‑2.B (LO), LIM‑2.B.1 (EK), LIM‑2.B.2 (EK)

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Continuity over an intervalFunctions continuous on all real numbersFunctions continuous at specific x-values

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Continuity over an intervalGet 3 of 4 questions to level up!

Continuity and common functionsGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑2 (EU), LIM‑2.C (LO), LIM‑2.C.1 (EK), LIM‑2.C.2 (EK)

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Removing discontinuities (factoring)Removing discontinuities (rationalization)

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Removable discontinuitiesGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑2 (EU), LIM‑2.D (LO), LIM‑2.D.1 (EK), LIM‑2.D.2 (EK)

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Introduction to infinite limitsInfinite limits and asymptotesAnalyzing unbounded limits: rational functionAnalyzing unbounded limits: mixed function

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Infinite limits: graphicalGet 3 of 4 questions to level up!

Infinite limits: algebraicGet 3 of 4 questions to level up!

AP Calc: LIM (BI), LIM‑2 (EU), LIM‑2.D (LO), LIM‑2.D.3 (EK), LIM‑2.D.4 (EK), LIM‑2.D.5 (EK)

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Introduction to limits at infinityFunctions with same limit at infinityLimits at infinity of quotients (Part 1)Limits at infinity of quotients (Part 2)Limits at infinity of quotients with square roots (odd power)Limits at infinity of quotients with square roots (even power)

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Limits at infinity: graphicalGet 3 of 4 questions to level up!

Limits at infinity of quotientsGet 5 of 7 questions to level up!

Limits at infinity of quotients with square rootsGet 3 of 4 questions to level up!

AP Calc: FUN (BI), FUN‑1 (EU), FUN‑1.A (LO), FUN‑1.A.1 (EK)

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Intermediate value theoremWorked example: using the intermediate value theoremJustification with the intermediate value theorem: tableJustification with the intermediate value theorem: equationIntermediate value theorem review

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Using the intermediate value theoremGet 3 of 4 questions to level up!

Justification with the intermediate value theoremGet 3 of 4 questions to level up!

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Formal definition of limits Part 1: intuition reviewFormal definition of limits Part 2: building the ideaFormal definition of limits Part 3: the definitionFormal definition of limits Part 4: using the definition

### About this unit

Limits describe the behavior of a function as we approach a certain input value, regardless of the function's actual value there. Continuity requires that the behavior of a function around a point matches the function's value at that point. These simple yet powerful ideas play a major role in all of calculus.AP® is a registered trademark of the College Board, which has not reviewed this resource.