Overview
This lecture covers testing the continuity of a piecewise function at ( x = -1 ) using the three-step continuity checklist.
Piecewise Function Definition
- The function ( f(x) ) is ( \frac{x^2 + x}{x + 1} ) when ( x \neq -1 ).
- For ( x = -1 ), ( f(-1) = 2 ).
Continuity Checklist
- 1: The function is defined at ( x = -1 ); ( f(-1) = 2 ).
- 2: The limit as ( x \to -1 ) of ( f(x) ) is found using ( \frac{x^2 + x}{x + 1} ).
- Substitute ( x = -1 ) to get ( 0/0 ); factor numerator to ( x(x + 1) ).
- After cancellation, limit becomes ( \lim_{x \to -1} x = -1 ).
- 3: For continuity at ( x = -1 ), require ( \lim_{x \to -1} f(x) = f(-1) ).
- The limit at ( x = -1 ) is ( -1 ), but ( f(-1) = 2 ); not equal.
Conclusion
- The function ( f ) is not continuous at ( x = -1 ) because the limit and value at that point are not equal.
- The function is continuous elsewhere except at ( x = -1 ).
Key Terms & Definitions
- Piecewise Function — a function defined by different formulas for different input values.
- Continuity Checklist — a three-step process to test if a function is continuous at a point: function defined, limit exists, limit equals function value.
- Removable Discontinuity — occurs when the limit exists but does not match the function value at that point.
Action Items / Next Steps
- Practice applying the continuity checklist to other piecewise functions.
- Review factoring methods for simplifying limits.