Question

question
find the horizontal asymptote for the function given below. give your answer in the form y = b, or enter ø if there is none.

f(x) = 7x^4 + 3x^2 + 1 / 3x^4 - 5x - 2
provide your answer below:

Explanation:

Step1: Determine the degrees of numerator and denominator

The degree of the numerator \(7x^{4}+3x^{2}+1\) is \(n = 4\). The degree of the denominator \(3x^{4}-5x - 2\) is \(m=4\).

Step2: Use the horizontal asymptote rule for \(n = m\)

When \(n = m\) (where \(n\) is the degree of the numerator and \(m\) is the degree of the denominator), the horizontal asymptote \(y\) is given by \(y=\frac{a_{n}}{b_{m}}\), where \(a_{n}\) is the leading coefficient of the numerator and \(b_{m}\) is the leading coefficient of the denominator. Here, \(a_{n}=7\) and \(b_{m}=3\).

Answer:

\(y = \frac{7}{3}\)