Question

Find the horizontal​ asymptote, if​ any, of the graph of the rational function.
​f(x)equalsStartFraction 19 x Over 5 x squared plus 6 EndFraction
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The horizontal asymptote is
  
enter your response here. ​(Type an​ equation.)
B.
There is no horizontal asymptote.

Answer 1

To find the horizontal asymptote of the rational function \( f(x) = \frac{19x}{5x^2 + 6} \), we compare the degrees of the polynomial in the numerator to the polynomial in the denominator.

1. **Degree of the numerator**: The numerator \( 19x \) has a degree of 1.
2. **Degree of the denominator**: The denominator \( 5x^2 + 6 \) has a degree of 2.

According to the rules for finding horizontal asymptotes:

- 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 equal to the degree of the denominator, the horizontal asymptote is given by the ratio of the leading coefficients.
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Since \( 1 < 2 \), we have:

The horizontal asymptote is \( y = 0 \).

So the correct choice is:

A. The horizontal asymptote is \( y = 0 \).