Learning Outcomes

- Understand the end behavior of polynomial functions.
- Find horizontal asymptotes and vertical asymptotes of rational functions.
- Sketch the graph of a polynomial or a rational function.

**Textbook **

- Chapter 4.6 Limits at Infinity and Asymptotes
*(this lesson is usually covered in two lectures)*

**Textbook Assignment**

- p. 436: 271, 273, 274, 279, 281, 298

**WeBWork Assignment**

- Limits-Infinite
- Application-Asymptotes
- Application-Shape of Graphs

**Exit problems ****of the session **

- Find the horizontal and vertical asymptotes of the following rational functions. (a). (b). .
- Sketch the graph of .
- Sketch the graph of .

** **

**Key Concepts**

- For a polynomial function , the end behavior is determined by the leading term . If , approaches or at each end.
- For a rational function , the end behavior is determined by the relationship between the degree of and the degree of :
- degree of the degree of : then the line is a horizontal asymptote for
- degree of the degree of : then the line is a horizontal asymptote, where and are the leading coefficients of and , respectively
- degree of the degree of : then approaches or at each end.

- When sketching a graph, you need all the information from Lesson 17.

#### Videos and Practice Problems of Selected Topics

**Limits at infinity and asymptotes**- Infinite limits and asymptotes (4:13) Using Desmos to analyze graphs of functions and analyze asymptotes using limits.
- Limits at infinity of rational functions (4:06) For , find and .

**Sketching the graph of a polynomial function**- Curve sketching with calculus: polynomial (20:30) Sketch the graph of using derivatives.
- Analyzing a function with its derivative (9:41) Sketch the graph of using derivatives.

**Sketching the graph of a rational function**- Curve sketching with calculus: rational function

Graph .