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The image asks to find the horizontal asymptotes of the given functions. Here’s how to determine the horizontal asymptotes for each function:

### 1. Horizontal Asymptote Rules:
- If the degrees of the numerator and denominator are the same, the horizontal asymptote is given by the ratio of the leading coefficients.
- 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 greater than the degree of the denominator, there is no horizontal asymptote (there may be an oblique asymptote instead).

### Finding the Horizontal Asymptotes:

1. **\( f(x) = \frac{2x^2 - 3}{5x^2 + 4} \)**
   - Degree of numerator = 2
   - Degree of denominator = 2
   - Leading coefficients ratio: \(\frac{2}{5}\)
   - **Horizontal Asymptote: \( y = \frac{2}{5} \)**

2. **\( f(x) = \frac{x^3 - 2x}{4x^3 - x + 1} \)**
   - Degree of numerator = 3
   - Degree of denominator = 3
   - Leading coefficients ratio: \(\frac{1}{4}\)
   - **Horizontal Asymptote: \( y = \frac{1}{4} \)**

3. **\( f(x) = \frac{7x^4 - x}{2x^2 + 3} \)**
   - Degree of numerator = 4
   - Degree of denominator = 2
   - Numerator degree > Denominator degree.
   - **Horizontal Asymptote: None (Oblique asymptote might exist).**

4. **\( f(x) = \frac{5x + 1}{x^2 - 2} \)**
   - Degree of numerator = 1
   - Degree of denominator = 2
   - Numerator degree < Denominator degree.
   - **Horizontal Asymptote: \( y = 0 \)**

Let me know if you need further details or clarifications! Here are some related questions:

1. How do you find oblique asymptotes when the degree of the numerator is greater than the degree of the denominator?
2. What happens if the degrees of the numerator and denominator are the same but the leading coefficients are zero?
3. How do vertical asymptotes differ from horizontal asymptotes?
4. Can a function have both horizontal and oblique asymptotes?
5. How do limits at infinity help determine horizontal asymptotes?

**Tip:** Always compare the degrees of the numerator and denominator to quickly determine the horizontal asymptote of a rational function.

Find the horizontal asymptotes of the given functions: f(x) = (2x^2 - 3) / (5x^2 + 4), f(x) = (x^3 - 2x) / (4x^3 - x + 1), f(x) = (7x^4 - x) / (2x^2 + 3), f(x) = (5x + 1) / (x^2 - 2)

Discover how to find the horizontal asymptotes for rational functions by comparing the degrees of the numerator and denominator. This solution involves key concepts in limits and asymptotes, suitable for students in Grades 11-12.

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