Overview
This lesson introduces the concept of limits in calculus, explaining how to determine the limit of a function as x approaches a certain point, regardless of the function's value at that point.
Understanding Limits with Graphs
- A limit describes what y-value a function approaches as x gets close to a specific value.
- Even if the function is not defined at that point, the limit can still exist by observing values on either side.
- The limit as x approaches 2 of a function can be 4, even if the function is not explicitly defined at x = 2.
- The graphical window analogy illustrates we can predict the limit by looking at the function near the hidden point.
Formal Definition of a Limit
- The limit of f(x) as x approaches a value (e.g., 2) equals L if f(x) gets as close as desired to L as x gets close to that value.
- It does not matter what, if anything, is happening exactly at the specific point.
Limit Example with a Function
- For f(x) = (x - 1)/(x^2 - 1), as x approaches 1, the function value heads toward 0.5, despite being undefined at x = 1.
- The rollercoaster or road analogy shows the function approaching a particular height (y-value) as x nears a certain value, regardless of a "pothole" at that point.
Key Terms & Definitions
- Limit — The value a function approaches as the input (x) gets close to a particular point.
- Undefined — When an expression or function does not have a meaningful value at a specific input (e.g., division by zero).
- Approaching — Getting arbitrarily close to a certain value from either side (left or right) on the graph.
Action Items / Next Steps
- Practice determining limits from graphs for points where the function might not be defined.
- Review and try calculating limits for simple rational functions.