AP Calculus BC

Advanced Placement Calculus BC including series, parametric equations, and polar functions.
Limits and ContinuityDifferentiation and ApplicationsIntegration and Its Uses
Infinite Series and ConvergenceParametric Equations and MotionPolar Coordinates and Graphs
Analyzing Population GrowthDesigning Roller Coasters with Parametric EquationsSignal Processing and Polar Graphs
Mastering Multiple-Choice QuestionsEffective Free-Response Techniques
Limits and Continuit...Differentiation and ...Integration and Its ...Infinite Series and ...Parametric Equations...Polar Coordinates an...Analyzing Population...Designing Roller Coa...Signal Processing an...Mastering Multiple-C...Effective Free-Respo...
Basic Concepts

Limits and Continuity

Understanding Limits

Limits are the foundation of calculus. They describe how a function behaves as its input approaches a particular value, even if the function isn't actually defined at that point.

  • If \( \lim_{x \to a} f(x) = L \), then as \( x \) gets closer to \( a \), \( f(x) \) gets closer to \( L \).
  • Limits help define derivatives and integrals.

Continuity

A function is continuous at a point if:

  1. The function is defined at the point.
  2. The limit exists at the point.
  3. The value of the function equals the limit at that point.

Why It Matters

The concept of limits allows us to work with functions that have jumps, holes, or even asymptotes, and is essential for understanding change.

Examples

  • The limit of \( f(x) = \frac{\sin x}{x} \) as \( x \) approaches 0 is 1.

  • A function with a hole at \( x = 2 \) is not continuous there.

In a Nutshell

Limits tell us what value a function approaches; continuity means no breaks in the graph.

Key Terms

Limit
The value a function approaches as the input approaches a certain point.
Continuity
A property where a function has no breaks, jumps, or holes at a point.
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Differentiation and Applications
AP Calculus BC Content & Lessons - Comprehensive Study Guide | Practice Hub