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 4: Detailed Asymptote Analysis

This page continues the comprehensive analysis of the function f(x) = (x² - 3) / (x² - 9), providing more detailed calculations and interpretations.

The analysis begins with a closer look at the vertical asymptote at x = 3. The page demonstrates how to calculate limits as x approaches 3 from both the left and right sides, confirming the existence of the vertical asymptote.

Example: lim[x→3⁺] (x² - 3) / (x² - 9) = +∞ and lim[x→3⁻] (x² - 3) / (x² - 9) = -∞

Next, the page focuses on finding horizontal asymptotes. It shows the calculation of the limit as x approaches positive and negative infinity, revealing that the function has a horizontal asymptote at y = 1.

Highlight: The existence of a horizontal asymptote precludes the possibility of an oblique asymptote for this function.

The position of the function's branches relative to the asymptotes is then determined by evaluating the function at very large positive and negative x values. This helps in understanding how the function behaves as x approaches infinity.

Example: f(100) ≈ 1.006 and f(-100) ≈ 1.006, indicating that the function approaches the horizontal asymptote y = 1 from above.

The page concludes with a summary of the asymptotic behavior of the function:

1. Vertical asymptote at x = 3
2. Horizontal asymptote at y = 1
3. No oblique asymptote

A graphical representation is provided to visualize how these asymptotes relate to the function's graph, emphasizing the importance of asymptote analysis in understanding function behavior.

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.