Overview
This lecture covers how to find limits and function values for piecewise-defined functions, and how to choose a constant for continuity.
Limits for Piecewise Functions
- A piecewise function defines different rules for different domains of x.
- To find the left-side limit at x = a, use the function rule valid for x < a.
- To find the right-side limit at x = a, use the rule for x > a or x ≥ a.
- The two-sided limit at x = a exists if both one-sided limits are equal.
Example 1: f(x) with Three Rules
- f(x) = 5x + 3 for x < 2; 2x² + 5 for 2 ≤ x < 4; x³ - 5x + 3 for x ≥ 4.
- Limit as x→2⁻: Use 5x + 3; plug in 2: 5*2 + 3 = 13.
- Limit as x→2⁺: Use 2x² + 5; plug in 2: 2*4 + 5 = 13.
- Both sided limits at x=2 equal 13, so limit as x→2 is 13.
- f(2): Use 2x ² + 5; f(2) = 13.
- Limit as x→4⁻: Use 2x² + 5; plug in 4: 2*16 + 5 = 37.
- Limit as x→4⁺: Use x³ - 5x + 3; plug in 4: 64 - 20 + 3 = 47.
- Limits at x=4 don't match, so limit as x→4 does not exist.
- f(4): Use x³ - 5x + 3; f(4) = 47.
Example 2: Another Piecewise Function
- f(x) = 7x - 5 for x < 1; f(x) = 5 for x = 1; f(x) = 3x² - x for 1 < x ≤ 3; f(x) = x³ + 4 for x > 3.
- Limit as x→1⁻: Use 7x - 5; plug in 1: 2.
- Limit as x→1⁺: Use 3x² - x; plug in 1: 2.
- Both sided limits at x=1 equal 2, so limit as x→1 is 2.
- f(1) = 5.
- Limit as x→3⁻: Use 3x² - x; plug in 3: 24.
- Limit as x→3⁺: Use x³ + 4; plug in 3: 31.
- Limits at x=3 don't match, so limit as x→3 does not exist.
- f(3) = 24.
Continuity and Solving for Constants
- To make f(x) = Cx + 3 for x < 2, and f(x) = 3x + C for x ≥ 2 continuous at x=2, set left and right expressions equal at x=2.
- Set 2C + 3 = 6 + C; solve for C: C = 3.
Key Terms & Definitions
- Piecewise Function — A function defined by different rules over different intervals of its domain.
- One-sided Limit — The value a function approaches from one direction (left or right) at a specific point.
- Continuity — A function is continuous at a point if its left and right limits and value at that point are all equal.
Action Items / Next Steps
- Practice finding one-sided and two-sided limits for additional piecewise functions.
- Review function continuity and how to solve for constants to ensure continuity.