Question

hw 3.7: rational functions
score: 6.5/28 answered: 7/28
question 8
find any horizontal asymptotes of the function.
$f(x) = \frac{1 - x^3 + 2x}{-5x^2 + 5x^3 - 3}$
question help: video message instructor

Explanation:

Step1: Identify degrees of numerator and denominator

First, we find the degree of the numerator and the denominator. For the numerator \(1 - x^{3}+2x\), the highest power of \(x\) is \(3\) (from \(-x^{3}\)), so the degree of the numerator \(n = 3\). For the denominator \(- 5x^{2}+5x^{3}-3\), the highest power of \(x\) is \(3\) (from \(5x^{3}\)), so the degree of the denominator \(m=3\).

Step2: Find leading coefficients

Next, we find the leading coefficients (the coefficients of the highest - degree terms). For the numerator, the leading term is \(-x^{3}\), so the leading coefficient \(a=- 1\). For the denominator, the leading term is \(5x^{3}\), so the leading coefficient \(b = 5\).

Step3: Apply the rule for horizontal asymptotes (when \(n = m\))

When the degree of the numerator \(n\) is equal to the degree of the denominator \(m\) (i.e., \(n=m\)), the horizontal asymptote \(y\) is given by the ratio of the leading coefficients of the numerator and the denominator. That is, \(y=\frac{a}{b}\).

Substituting \(a=-1\) and \(b = 5\) into the formula, we get \(y=\frac{-1}{5}=-\frac{1}{5}\).

Answer:

The horizontal asymptote of the function \(f(x)\) is \(y =-\frac{1}{5}\)