Math / Calculus

QuestionDetermine the horizontal asymptote of the function. If none exists, state that fact. $f(x)=\frac{2 x^{3}-4 x+5}{6 x^{3}+9 x-5}$
Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. A. The function has one horizontal asymptote, $\square$ (Type an equation.) B. The function has two horizontal asymptotes. The top asymptote is $\square$ and the bottom asymptote is $\square$ (Type equations.) C. The function has no horizontal asymptotes.

Studdy Solution

STEP 1

1. A horizontal asymptote of a rational function $f(x) = \frac{P(x)}{Q(x)}$ is determined by the degrees of the polynomials $P(x)$ and $Q(x)$ .
2. If the degree of $P(x)$ is less than the degree of $Q(x)$ , the horizontal asymptote is $y = 0$ .
3. If the degree of $P(x)$ is equal to the degree of $Q(x)$ , the horizontal asymptote is $y = \frac{a}{b}$ , where $a$ and $b$ are the leading coefficients of $P(x)$ and $Q(x)$ , respectively.
4. If the degree of $P(x)$ is greater than the degree of $Q(x)$ , there is no horizontal asymptote.

STEP 2

1. Identify the degrees of the numerator and the denominator.
2. Compare the degrees to determine the presence and equation of a horizontal asymptote.

STEP 3

Identify the degrees of the numerator and the denominator:
The numerator is $2x^3 - 4x + 5$ , and its degree is 3. The denominator is $6x^3 + 9x - 5$ , and its degree is 3.

STEP 4

Compare the degrees of the numerator and the denominator:
Since the degree of the numerator (3) is equal to the degree of the denominator (3), we use the leading coefficients to determine the horizontal asymptote.

STEP 5

Find the leading coefficients:
The leading coefficient of the numerator is 2. The leading coefficient of the denominator is 6.
The horizontal asymptote is given by:
$y = \frac{2}{6} = \frac{1}{3}$
The function has one horizontal asymptote, $y = \frac{1}{3}$ .

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