Finding Horizontal and Vertical Asymptotes - Part 2 (01-10-14)

Today's class focused on understanding and identifying horizontal and vertical asymptotes of rational functions. Asymptotes help us analyze the end behavior of functions and identify points where the function is undefined.

Key Concepts

  • Vertical Asymptotes (V.A.): Occur where the denominator of a rational function equals zero, making the function undefined. To find them, set the denominator equal to zero and solve for $x$. For example, in the function $f(x) = \frac{1}{x}$, the vertical asymptote is $x = 0$.
  • Horizontal Asymptotes (H.A.): Describe the behavior of the function as $x$ approaches positive or negative infinity. To find them, compare the degrees of the numerator and denominator:
    • If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.
    • If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is $y = \frac{\text{leading coefficient of numerator}}{\text{leading coefficient of denominator}}$.
    • If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (there may be a slant/oblique asymptote).

Examples

Let's look at some examples to illustrate these concepts:

  1. Example 1: $f(x) = \frac{1}{x^2}$
    Domain: $x \neq 0$
    V.A.: $x = 0$
    H.A.: $y = 0$
  2. Example 2: $f(x) = \frac{3}{(x-2)^3}$
    Domain: $x \neq 2$
    V.A.: $x = 2$
    H.A.: $y = 0$
  3. Example 3: $f(x) = \frac{x^2 - 25}{x^2 + 5x}$
    $f(x) = \frac{(x+5)(x-5)}{x(x+5)} = \frac{x-5}{x}$ (after simplification, noting the hole at x=-5)
    Domain: $x \neq 0, x \neq -5$
    V.A.: $x = 0$
    Hole: $x=-5$
    H.A.: $y = 1$
  4. Example 4: $f(x) = \frac{3x^2 + x - 5}{x^2 + 1}$
    Domain: All real numbers ($\mathbb{R}$)
    V.A.: None
    H.A.: $y = 3$
  5. Example 5: $f(x) = \frac{x-3}{|x|}$
    When x>0, f(x) = $\frac{x-3}{x}$
    When x<0, f(x) = $\frac{x-3}{-x}$
    Domain: $x \neq 0$
    V.A.: $x = 0$
    H.A.: $y = 1, y = -1$
  6. Example 6: $f(x) = 3 - \frac{2}{x-4}$
    $f(x) = \frac{3x-14}{x-4}$
    Domain: $x \neq 4$
    V.A.: $x = 4$
    H.A.: $y = 3$

Homework

Practice these concepts! Complete the following problems:

  • Page 149-151: #31-34, 44, 46

Keep practicing, and you'll master finding asymptotes in no time! Remember to always check the domain first and simplify the rational function when possible.