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). $y=\dfrac{1}{x^2+x^3}$ (b). $y=\dfrac{x^2-x^3}{x+2x^3}$.
- Sketch the graph of $f(x)=4x^3-3x^2$.
- Sketch the graph of $f(x)=\dfrac{x}{x^2-4}$.

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**Key Concepts**

- For a polynomial function $p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, the end behavior is determined by the leading term $a_nx^n$. If $n\neq 0$, $p(x)$ approaches $\infty$ or $-\infty$ at each end.
- For a rational function $f(x)=\dfrac{p(x)}{q(x)}$, the end behavior is determined by the relationship between the degree of $p(x)$ and the degree of $q(x)$:
- degree of $p(x)$ $<$ the degree of $q(x)$: then the line $y=0$ is a horizontal asymptote for $f(x)$
- degree of $p(x)$ $=$ the degree of $q(x)$: then the line $y=\dfrac{a_n}{b_n}$ is a horizontal asymptote, where $a_n$ and $b_n$ are the leading coefficients of $p(x)$ and $q(x)$, respectively
- degree of $p(x)$ $>$ the degree of $q(x)$: then $f(x)$ approaches $\infty$ or $-\infty$ 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**- $\rhd$ Infinite limits and asymptotes (4:13) Using Desmos to analyze graphs of functions and analyze asymptotes using limits.
- $\rhd$ Limits at infinity of rational functions (4:06) For $f(x) = \dfrac{4x^5-3x^2+3}{6x^5-100x^2-10}$, find $\displaystyle\lim_{x\to\infty}f(x)$ and $\displaystyle\lim_{x\to-\infty}f(x)$.

**Sketching the graph of a polynomial function**- $\rhd$ Curve sketching with calculus: polynomial (20:30) Sketch the graph of $f(x) = 3x^4 -4x^3+2$ using derivatives.
- $\rhd$ Analyzing a function with its derivative (9:41) Sketch the graph of $f(x)=x^3-12x-2$ using derivatives.

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

Graph $y=\dfrac{x-1}{x^2}$.