Find the horizontal asymptote of g(x) = ((2x^2)-(3^x))/((4^x)-(5x^2))

To find the horizontal asymptote of the function \( g(x) = \frac{2x^2 - 3^x}{4^x - 5x^2} \), we need to analyze the behavior of the function as \( x \) approaches infinity (\( x \to \infty \)) or negative infinity (\( x \to -\infty \)).

### Step 1: Analyze the growth rates of the terms
- \( 2x^2 \) grows quadratically.
- \( 3^x \) grows exponentially.
- \( 4^x \) grows exponentially.
- \( 5x^2 \) grows quadratically.

Since exponential functions grow much faster than polynomial functions as \( x \) becomes large, the terms involving \( 3^x \) and \( 4^x \) will dominate as \( x \to \infty \).

### Step 2: Simplify the function for large \( x \)
For large \( x \), we can approximate \( g(x) \) by considering only the leading exponential terms:
\[
g(x) \approx \frac{-3^x}{4^x} = \frac{-\left(3/4\right)^x}{1}
\]
As \( x \to \infty \), the term \( \left(\frac{3}{4}\right)^x \) approaches 0 because \( \frac{3}{4} < 1 \). Therefore:
\[
g(x) \to 0 \text{ as } x \to \infty
\]

### Step 3: Conclusion
The horizontal asymptote of the function \( g(x) \) is:
\[
y = 0
\]

Would you like more details or have any questions?

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Here are 5 related questions:
1. What is the vertical asymptote of \( g(x) \)?
2. How do the horizontal asymptotes change if the exponents are altered?
3. Can we find the horizontal asymptote if \( g(x) \) had logarithmic functions?
4. What happens to the asymptotes if the coefficients of \( x^2 \) terms change?
5. How do we determine slant (oblique) asymptotes?

**Tip:** When finding horizontal asymptotes, focus on the terms that grow the fastest as \( x \) approaches infinity or negative infinity.

Learn how to find the horizontal asymptote of the function g(x) = ((2x^2)-(3^x))/((4^x)-(5x^2)) using limits and exponential function analysis. Understand the behavior of functions at infinity and how exponential growth affects asymptotic behavior. Suitable for college-level mathematics.

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