QuestionFind the horizontal asymptote, if any, of the graph of the rational function. $f(x)=\frac{17 x}{7 x^{2}+5}$
Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The horizontal asymptote is $\square$ . (Type an equation.) B. There is no horizontal asymptote.

Studdy Solution

STEP 1

1. The function $f(x) = \frac{17x}{7x^2 + 5}$ is a rational function.
2. Horizontal asymptotes are determined by the degrees of the numerator and the denominator.

STEP 2

1. Identify the degrees of the numerator and the denominator.
2. Compare the degrees to determine the existence of a horizontal asymptote.
3. Determine the equation of the horizontal asymptote if it exists.

STEP 3

Identify the degrees of the numerator and the denominator:
- The degree of the numerator $17x$ is 1. - The degree of the denominator $7x^2 + 5$ is 2.

STEP 4

Compare the degrees of the numerator and the denominator:
- 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 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.
In this case, the degree of the numerator (1) is less than the degree of the denominator (2).

STEP 5

Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is:
$y = 0$
The horizontal asymptote is $\boxed{y = 0}$ .

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