Page 2: Calculating Asymptotes

This page delves deeper into the methods for calculating different types of asymptotes. It focuses on the use of limits to determine the existence and equations of asymptotes.

For vertical asymptotes, the page explains how to find them by looking at values of x that make the denominator of a rational function zero. It provides a step-by-step approach to calculating limits as x approaches these critical values from both sides.

Example: For f(x) = (x³ + 8) / (x² - 4), we calculate the limit as x approaches 2 from both sides to confirm the vertical asymptote at x = 2.

The page then moves on to horizontal asymptotes, explaining how to calculate them by evaluating the limit of the function as x approaches positive or negative infinity. It emphasizes that when these limits exist and are finite, they represent horizontal asymptotes.

Vocabulary: A horizontal asymptote is a line y = k where k is the limit of the function as x approaches infinity.

The relationship between the degrees of the numerator and denominator in rational functions is discussed, as it determines the behavior of the function at infinity and thus the existence of horizontal asymptotes.

Highlight: When a function has a horizontal asymptote, it cannot have an oblique asymptote.

The page concludes with examples of how to determine the position of function branches relative to asymptotes, which is crucial for accurate graphing.

Page 6: Advanced Asymptote Problems

This page presents a more complex asymptote problem, demonstrating advanced techniques for asymptote analysis.

The function analyzed is f(x) = (x² - 1) / (x² - 2)

The analysis includes:

1. Identifying vertical asymptotes:

   • Solving x² - 2 = 0 yields x = ±√2
   • Confirming vertical asymptotes at x = √2 and x = -√2 through limit calculations

2. Calculating limits for x approaching √2 from both sides:

   • lim[x→√2⁺] f(x) = +∞
   • lim[x→√2⁻] f(x) = -∞

3. Checking for horizontal asymptotes:

   • lim[x→±∞] (x² - 1) / (x² - 2) = 1
   • Confirming a horizontal asymptote at y = 1

4. Determining the position of function branches relative to asymptotes:

   • Evaluating f(100) and f(-100) to understand behavior at large x values

Example: f(100) ≈ 1.0001 and f(-100) ≈ 1.0001, indicating that the function approaches the horizontal asymptote y = 1 from above.

5. Confirming the non-existence of oblique asymptotes due to the presence of a horizontal asymptote

Highlight: The function has vertical asymptotes at x = ±√2 and a horizontal asymptote at y = 1, with no oblique asymptote.

The page also includes a brief discussion on the domain of the function: D = ℝ - {±√2}

This advanced example showcases the application of limit calculations and asymptote analysis techniques to a more complex rational function, reinforcing the concepts learned in previous sections.

Vocabulary: Rational function - a function that can be expressed as the ratio of two polynomials.

The comprehensive analysis provided in this example serves as an excellent model for approaching complex asymptote problems in advanced mathematics courses.

Page 7: Special Cases

Addresses special cases and unique situations in asymptote calculation.

Definition: Special attention to cases where horizontal asymptotes preclude the existence of oblique asymptotes.

Example: Analysis of the function 5/(x²-2x) with multiple vertical asymptotes.