Find the vertical asymptote(s) and determine the infinite limit(s) of f(x).                            f (x) =  x² - 4x  +3  /   x²  - 3x + 2

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Chapter1: Functions And Models
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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Find the vertical asymptote(s) and determine the infinite limit(s) of f(x).

                           f (x) =  x² - 4x  +3  /   x²  - 3x + 2 

### Vertical Asymptotes and Infinite Limits of Rational Functions

**Problem Statement:**
Find the vertical asymptote(s) and determine the infinite limit(s) of \( f(x) \).

\[ f(x) = \frac{x^2 - 4x + 3}{x^2 - 3x + 2} \]

**Explanation:**
This problem involves analyzing the function \( f(x) \) to find its vertical asymptotes and evaluate its behavior as \( x \) approaches these points. 

**Steps to Solve:**

1. **Factor the Numerator and Denominator:**
   - Numerator: \( x^2 - 4x + 3 \)
   - Denominator: \( x^2 - 3x + 2 \)

2. **Find the Roots of the Denominator:**
   Set \( x^2 - 3x + 2 = 0 \) to find the values where the denominator is zero, which are the potential locations of vertical asymptotes.

3. **Simplification (if possible):**
   Check if there’s common factoring in the numerator and denominator that could simplify the function.

4. **Determine Vertical Asymptotes:**
   Identify the values of \( x \) that make the denominator zero but do not make the numerator zero after simplification, which will help confirm the vertical asymptotes.

5. **Evaluate Infinite Limits:**
   Determine the behavior of \( f(x) \) as \( x \) approaches the vertical asymptotes.

**Graphical Interpretation:**
Include a graph of \( f(x) \) as an additional resource to visually confirm the locations of vertical asymptotes and observe the function's behavior near these points.

By following these steps, you can systematically determine the vertical asymptotes and infinite limits of the given function.
Transcribed Image Text:### Vertical Asymptotes and Infinite Limits of Rational Functions **Problem Statement:** Find the vertical asymptote(s) and determine the infinite limit(s) of \( f(x) \). \[ f(x) = \frac{x^2 - 4x + 3}{x^2 - 3x + 2} \] **Explanation:** This problem involves analyzing the function \( f(x) \) to find its vertical asymptotes and evaluate its behavior as \( x \) approaches these points. **Steps to Solve:** 1. **Factor the Numerator and Denominator:** - Numerator: \( x^2 - 4x + 3 \) - Denominator: \( x^2 - 3x + 2 \) 2. **Find the Roots of the Denominator:** Set \( x^2 - 3x + 2 = 0 \) to find the values where the denominator is zero, which are the potential locations of vertical asymptotes. 3. **Simplification (if possible):** Check if there’s common factoring in the numerator and denominator that could simplify the function. 4. **Determine Vertical Asymptotes:** Identify the values of \( x \) that make the denominator zero but do not make the numerator zero after simplification, which will help confirm the vertical asymptotes. 5. **Evaluate Infinite Limits:** Determine the behavior of \( f(x) \) as \( x \) approaches the vertical asymptotes. **Graphical Interpretation:** Include a graph of \( f(x) \) as an additional resource to visually confirm the locations of vertical asymptotes and observe the function's behavior near these points. By following these steps, you can systematically determine the vertical asymptotes and infinite limits of the given function.
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