Overview
This lecture introduces finite or asymptotic end behavior limits, focusing on horizontal asymptotes, their interpretation, and how to determine them from graphs.
Definitions and Interpretations
- A horizontal asymptote is a line ( y = l ) that the function ( f(x) ) approaches as ( x ) goes to infinity or negative infinity.
- The limit statement for right-side end behavior is: ( \lim_{x \to \infty} f(x) = l ), meaning ( f(x) ) approaches ( l ) as ( x ) increases without bound.
- The graph can approach the asymptote from above or below; the direction does not change the limit.
- The limit statement for left-side end behavior is: ( \lim_{x \to -\infty} f(x) = l ), meaning ( f(x) ) approaches ( l ) as ( x ) decreases without bound.
- The function may cross the asymptote before showing asymptotic behavior.
Interpreting Two-Sided Infinite Limits
- A two-sided infinite limit is read as ( f(x) ) approaches positive or negative infinity as ( x ) approaches ( a ) from both sides.
- To interpret, mark the vertical asymptote at ( x = a ) and show the function's behavior near this line.
Example: Finding Asymptotic End Behavior
- To find the finite end behavior limit as ( x \to -\infty ), identify the horizontal asymptote on the left.
- If the graph approaches ( y = -1 ) as ( x \to -\infty ), then ( \lim_{x \to -\infty} f(x) = -1 ).
- To find the finite end behavior limit as ( x \to \infty ), identify the horizontal asymptote on the right.
- If the graph approaches ( y = 2 ) as ( x \to \infty ), then ( \lim_{x \to \infty} f(x) = 2 ).
Key Terms & Definitions
- Horizontal Asymptote — A line ( y = l ) that the function approaches as ( x ) goes to ( \pm\infty ).
- Finite (Asymptotic) End Behavior Limit — The value ( l ) that ( f(x) ) approaches as ( x ) goes to ( \infty ) or ( -\infty ).
- Two-Sided Infinite Limit — Describes behavior as the function approaches infinity near a vertical asymptote ( x = a ).
Action Items / Next Steps
- Practice identifying horizontal asymptotes and calculating end behavior limits from graphs.
- Review related textbook sections on limits and asymptotes.