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Limits and continuity

Limits intro

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Limits intro
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Limits intro
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Limits introGet 3 of 4 questions to level up!

Estimating limits from graphs

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Estimating limit values from graphs
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Unbounded limits
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Estimating limit values from graphs
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One-sided limits from graphs
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One-sided limits from graphs: asymptote
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Connecting limits and graphical behavior
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Connecting limits and graphical behavior (more examples)
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Practice
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!

Estimating limits from tables

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Approximating limits using tables
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Estimating limits from tables
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Using tables to approximate limit values
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One-sided limits from tables
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Practice
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!

Formal definition of limits (epsilon-delta)

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Formal definition of limits Part 1: intuition review
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Formal definition of limits Part 2: building the idea
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Formal definition of limits Part 3: the definition
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Formal definition of limits Part 4: using the definition
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Properties of limits

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Limit properties
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Limits of combined functions
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Limits of combined functions: piecewise functions
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Limits of composite functions
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Practice
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!

Limits by direct substitution

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Limits by direct substitution
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Undefined limits by direct substitution
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Limits of trigonometric functions
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Limits of piecewise functions
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Limits of piecewise functions: absolute value
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Practice
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!

Limits using algebraic manipulation

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Limits by factoring
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Limits by rationalizing
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Trig limit using Pythagorean identity
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Trig limit using double angle identity
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Practice
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!

Strategy in finding limits

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Strategy in finding limits
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Strategy 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!

Squeeze theorem

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Squeeze theorem intro
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Limit of sin(x)/x as x approaches 0
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Limit of (1-cos(x))/x as x approaches 0
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Squeeze theoremGet 3 of 4 questions to level up!

Types of discontinuities

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Types of discontinuities
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Classify discontinuitiesGet 3 of 4 questions to level up!

Continuity at a point

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

Continuity over an interval

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Continuity over an interval
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Functions continuous on all real numbers
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Functions 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!

Removing discontinuities

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Removing discontinuities (factoring)
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Removing discontinuities (rationalization)
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Removable discontinuitiesGet 3 of 4 questions to level up!

Infinite limits

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Introduction to infinite limits
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Infinite limits and asymptotes
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Connecting limits at infinity notation and graph
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Analyzing unbounded limits: rational function
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Analyzing unbounded limits: mixed function
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Practice
Infinite limits: graphicalGet 3 of 4 questions to level up!
Infinite limits: algebraicGet 3 of 4 questions to level up!

Limits at infinity

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Introduction to limits at infinity
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Functions with same limit at infinity
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Limits at infinity of quotients (Part 1)
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Limits at infinity of quotients (Part 2)
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Limits at infinity of quotients with square roots (odd power)
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Limits at infinity of quotients with square roots (even power)
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Limits at infinity of quotients with trig
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Limits at infinity of quotients with trig (limit undefined)
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Limit at infinity of a difference of functions
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Practice
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!
Limits at infinity of quotients with trigGet 3 of 4 questions to level up!

Intermediate value theorem

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Intermediate value theorem
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Worked example: using the intermediate value theorem
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Justification with the intermediate value theorem: table
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Justification with the intermediate value theorem: equation
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Intermediate value theorem review
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Practice
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!

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.