Learning Outcomes

  1. Understand the end behavior of polynomial functions.
  2. Find horizontal asymptotes and vertical asymptotes of rational functions. 
  3. 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 

  1. Find the horizontal and vertical asymptotes of the following rational functions.                        (a).   $y=\dfrac{1}{x^2+x^3}$  (b).    $y=\dfrac{x^2-x^3}{x+2x^3}$.  
  2. Sketch the graph of  $f(x)=4x^3-3x^2$.
  3. Sketch the graph of  $f(x)=\dfrac{x}{x^2-4}$.

 

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
    1. $\rhd$ Infinite limits and asymptotes (4:13) Using Desmos to analyze graphs of functions and analyze asymptotes using limits.
    2. $\rhd$ 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 rational function
    1.  Curve sketching with calculus: rational function

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

      1. $\rhd$ Part I (10:00): Find the domain, intercepts, symmetry relations, asymptotes, intervals of increase/decrease, and local extrema.
      2. $\rhd$ Part II (8:04): Find the intervals of concavity and skech the graph of the function.