Identify the horizontal asymptote of f(x) = 3 over 5 x.

To identify the horizontal asymptote of the function \( f(x) = \frac{3}{5x} \), follow these steps:

1. **Express the Function**: 
   \[
   f(x) = \frac{3}{5x}
   \]

2. **Analyze the Behavior as \( x \) Approaches Infinity**: 
   - As \( x \) approaches infinity (\( x \to \infty \)), the term \( \frac{3}{5x} \) becomes smaller and smaller, approaching zero.

3. **Horizontal Asymptote**:
   - Therefore, as \( x \) approaches infinity or negative infinity, \( f(x) \) approaches 0.

The horizontal asymptote of \( f(x) = \frac{3}{5x} \) is \( y = 0 \).

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Here are some related questions:
1. What is the horizontal asymptote of \( f(x) = \frac{2}{x} \)?
2. How does the value of the numerator affect the horizontal asymptote in rational functions?
3. What happens to the horizontal asymptote if the denominator has a higher degree than the numerator?
4. Identify the vertical asymptote for \( f(x) = \frac{3}{5x} \).
5. What is the horizontal asymptote for \( f(x) = \frac{5x^2 + 3}{7x^2 - 2} \)?
6. How do you find horizontal asymptotes for exponential functions?
7. Explain how to find the horizontal asymptote of \( f(x) = \frac{3}{5x + 2} \).
8. Compare the horizontal asymptotes of \( f(x) = \frac{3}{5x} \) and \( g(x) = \frac{5x}{3} \).

**Tip**: Horizontal asymptotes show the behavior of a function as \( x \) approaches infinity or negative infinity; they're key in understanding the long-term behavior of the function.

Learn how to identify the horizontal asymptote of the rational function f(x) = 3/(5x). Understand the behavior of the function as x approaches infinity and its implication on long-term behavior. Suitable for high school students studying limits and rational functions.

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