AP Calculus Limit Examples

Introduction to Limits

• Definition of a limit: A limit exists if both one-sided limits exist.
• Focus on evaluating limits and determining when they do not exist.

Example 1: Limit as X Approaches 1

• Function: f(x) =
  • Cosine(pi * x) for x >= 1
  • ln(e * x) for x < 1
• One-sided limits:
  • Right Limit (x approaches 1 from positive side):
    • f(1) = cos(pi * 1) = -1
  • Left Limit (x approaches 1 from negative side):
    • f(1) = ln(e * 1) = 1
• Conclusion: Limit does not exist (DNE) because:
  • Right limit ≠ Left limit

Example 2: Limits as X Approaches 2

• One-sided limits:
  • Right Limit:
    • f(2) = sin(2pi/3)
    • 2pi/3 is in the second quadrant, sin(2pi/3) = √3/2
  • Left Limit:
    • f(2) = e^(2 - 2) = e^0 = 1

Limit Properties

• If the limit of f(x) as x approaches a is L, then:
  • Limit of 1/f(x) as x approaches a = 1/L (if L ≠ 0)

Example: Limit as X Approaches 0

• Limit of x/sin(4x) = 1/4
• Limit of 3x/tan(4x) can be simplified using trigonometric identities.

Continuity and Discontinuity

• Types of continuity:
  • Limit exists, but function value does not exist.
  • Limit exists and equals the function value.

Example 3: Composite Functions

• Example: G(G(x)) as x approaches 2
  • Analyze inner limit G(x) as x approaches 2.
  • G approaches 0 from above, leading to G(0), which does not exist, hence G(G(x)) does not exist.

Graphing Functions and Limits

• Example of a function f(x) approaching a limit:
  • f(1) = 4, limit as x approaches 1 = 5
• Identify functions:
  • Continuous: limit exists and equals function value.
  • Discontinuous: limit exists but does not equal function value.

Algebraic Manipulations

• Techniques for solving exponential equations (using natural logs)
• Simplifying expressions using exponent rules (i.e. x^n, powers of x)

Example: Limit as X Approaches 4

• Visually determine the limits from the graph:
  • Right limit, negative limit, and function value analysis.

Conclusion

• Importance of understanding limits, continuity, and algebraic manipulations for succeeding in AP Calculus.
• Explore further practice problems for deeper understanding.