Question
Find the horizontal asymptote, if any, of the graph of the rational function \[ h(x)=\frac{17 x^{3}}{2 x^{2}+9} \] Select the correct choice below and, if necessary, fill in the answer box to complete your chorce A. The horizontal asymptote is . (Type an equation) B. There is no horizontal asymptote A
Ask by Allan Ellis. in the United States
Feb 26,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
There is no horizontal asymptote.
Solution
To find the horizontal asymptote of the rational function
\[
h(x) = \frac{17 x^{3}}{2 x^{2} + 9},
\]
we need to analyze the degrees of the polynomial in the numerator and the polynomial in the denominator.
1. **Identify the degrees:**
- The degree of the numerator \(17 x^{3}\) is 3.
- The degree of the denominator \(2 x^{2} + 9\) is 2.
2. **Determine the horizontal asymptote based on the degrees:**
- If the degree of the numerator is greater than the degree of the denominator (which is the case here, since 3 > 2), then there is no horizontal asymptote.
Thus, the correct choice is:
B. There is no horizontal asymptote.
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Simplify this solution Beyond the Answer
To find the horizontal asymptote of the rational function \( h(x) = \frac{17 x^{3}}{2 x^{2} + 9} \), we need to compare the degrees of the polynomial in the numerator with the degree of the polynomial in the denominator. The degree of the numerator is 3 (from \( 17 x^3 \)) and the degree of the denominator is 2 (from \( 2 x^2 \)). When the degree of the numerator is greater than the degree of the denominator, as it is here, the horizontal asymptote does not exist. So the correct choice is: B. There is no horizontal asymptote.
