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 3: Oblique Asymptotes and Comprehensive Analysis

This page focuses on oblique asymptotes and provides a comprehensive example of analyzing all types of asymptotes for a given function.

The method for finding oblique asymptotes is explained in detail. It involves calculating two limits: one to determine the slope (m) and another to find the y-intercept (n) of the asymptote equation y = mx + n.

Definition: An oblique asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator in a rational function.

The page provides a step-by-step guide to calculating these limits and determining the equation of the oblique asymptote. It also explains how to check the position of the function relative to the asymptote.

Example: For the function f(x) = (x³ + 8) / (x² - 4), the oblique asymptote is calculated to be y = x + 0.

A comprehensive example is then presented, analyzing all types of asymptotes for the function f(x) = (x² - 3) / (x² - 9). This example demonstrates how to:

1. Find the domain of the function
2. Identify vertical asymptotes
3. Calculate limits to determine horizontal asymptotes
4. Check for oblique asymptotes
5. Determine the position of function branches relative to asymptotes

Highlight: A thorough asymptote analysis involves checking for all types of asymptotes and understanding how the function behaves around these asymptotes.

The page concludes with a graphical representation of the analyzed function, showing how the asymptotes relate to the function's graph.

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 5: Solved Exercises on Asymptotes

This page presents two solved exercises demonstrating the process of finding asymptotes for different functions.

Exercise 1: f(x) = (2x + 3) / (4x - 8)

The analysis for this function includes:

1. Finding the domain: D(f) = ℝ - {2}
2. Identifying the vertical asymptote at x = 2
3. Calculating limits to confirm the vertical asymptote
4. Determining the horizontal asymptote by evaluating limits as x approaches infinity
5. Checking the position of function branches relative to the asymptote

Example: The horizontal asymptote is found to be y = 1/2 by calculating lim[x→±∞] (2x + 3) / (4x - 8) = 1/2

Exercise 2: f(x) = x² / (x - 4)

This exercise follows a similar process:

1. Finding the domain: D(f) = ℝ - {4}
2. Identifying the vertical asymptote at x = 4
3. Calculating limits to confirm the vertical asymptote
4. Checking for a horizontal asymptote (none exists in this case)
5. Determining the existence of an oblique asymptote

Highlight: This function has an oblique asymptote y = x + 4, as the degree of the numerator is one more than the degree of the denominator.

The page emphasizes the importance of checking the position of function branches relative to asymptotes for accurate graphing.

Vocabulary: Oblique asymptote - an asymptote that is neither horizontal nor vertical, typically of the form y = mx + n.

These exercises provide practical applications of the asymptote concepts and techniques discussed in previous pages.

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 6: Complex Function Analysis

Focuses on more complex functions and their asymptotic behavior.

Highlight: Special attention to functions with square roots and their asymptotic behavior.

Example: Detailed analysis of the function (x²-1)/(x²-2) including all types of asymptotes.

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.

Page 8: Advanced Oblique Asymptotes

Detailed coverage of oblique asymptote calculations and special cases.

Formula: Advanced techniques for calculating oblique asymptotes using limits and algebraic manipulation.

Example: Complex function analysis with both vertical and oblique asymptotes.

Page 1: Introduction to Asymptotes and Discontinuity

This page introduces the concept of asymptotes and their types. It explains that asymptotes are lines that a function approaches but never reaches. The three main types of asymptotes are discussed: vertical, horizontal, and oblique.

Definition: An asymptote is a line that a graph of a function approaches as the x or y values get very large or very small.

The page also provides visual representations of each type of asymptote, showing how functions behave near these lines. It emphasizes the importance of understanding domain restrictions and limits when dealing with asymptotes.

Example: For the function f(x) = (x³ + 8) / (x² - 4), there is a vertical asymptote at x = 2.

The concept of discontinuity is introduced, highlighting its relationship with asymptotes. The page explains that discontinuities often occur at points where the function is undefined, which can lead to vertical asymptotes.

Highlight: Asymptotes help us understand the behavior of functions as x approaches infinity or specific values where the function is undefined.