To find the horizontal asymptote of the function \( G(x) = \frac{2x^2}{x^3 - 5} \), we analyze the degrees of the numerator and denominator.

### Step-by-Step Process:

1. **Identify the degrees of the numerator and denominator:**
   - The numerator \( 2x^2 \) is of degree 2.
   - The denominator \( x^3 - 5 \) is of degree 3.

2. **Compare the degrees:**
   - The degree of the numerator (2) is less than the degree of the denominator (3).

3. **Conclusion about the horizontal asymptote:**
   - When the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is \( y = 0 \).

So, the horizontal asymptote of \( G(x) \) is \( y = 0 \).

Would you like further details or explanations on any part of this process?

Here are 5 related questions that expand on this topic:

1. What are the vertical asymptotes of the function \( G(x) \)?
2. How do you find the horizontal asymptote if the degrees of the numerator and denominator are equal?
3. What is the behavior of the function \( G(x) \) as \( x \) approaches infinity?
4. Can you explain how to find oblique asymptotes in rational functions?
5. How do horizontal asymptotes differ from vertical asymptotes in terms of function behavior?

**Tip:** When analyzing rational functions for asymptotes, always compare the degrees of the numerator and denominator first. This gives you quick insights into the horizontal or slant asymptotes.

Discover how to find the horizontal asymptote of the rational function G(x) = 2x^2/(x^3 - 5). Learn step-by-step with explanations on comparing polynomial degrees and determining asymptotic behavior. Suitable for advanced high school students and college-level mathematics.

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