Overview
This lecture summarized the definitions of limits and explained how to use a graphing approach to determine limits of functions.
Definitions of Limits
- To find a limit, analyze the function as x approaches a target value.
- A graphing device can be used to visualize the function and identify limit behaviors.
- Limits are considered from the left, right, both sides, as x approaches negative infinity, and as x approaches positive infinity.
- If none of the recognizable behaviors fit, the limit is said not to exist.
The Graphing Approach
- Graph the function using a calculator or graphing tool.
- Identify the target value (a) for x and observe the function's behavior near this value.
- For each direction (left, right, both sides, negative infinity, positive infinity), check for three common behaviors.
- If the function matches one of the familiar patterns, describe it using the language of limits.
- If the function's behavior does not match known patterns, conclude the limit does not exist.
Common Limit Behaviors (per direction)
- The three behaviors for each direction typically include approaching a finite value, approaching infinity, or oscillating without settling.
- This pattern repeats whether x approaches from the left, right, both sides, or towards infinity.
Key Terms & Definitions
- Limit — The value a function approaches as the input (x) gets close to a target value.
- Graphing approach — Using a visual graph to determine the behavior of a function near a specific point.
Action Items / Next Steps
- Practice using a graphing calculator or tool to graph functions and observe limit behaviors.
- Review the 15 graphing definitions of limits for different directions and behaviors.