Question

find the horizontal asymptote, if any, of the graph of the rational function. g(x) = \frac{12x^{2}}{4x^{2}+5} select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the horizontal asymptote is. (type an equation.) b. there is no horizontal asymptote.

Explanation:

Step1: Identify degrees of polynomials

The degree of the numerator $12x^{2}$ is $n = 2$, and the degree of the denominator $4x^{2}+5$ is $m = 2$.

Step2: Apply horizontal - asymptote rule

When $n = m$, the horizontal asymptote is $y=\frac{a_{n}}{b_{m}}$, where $a_{n}$ is the leading - coefficient of the numerator and $b_{m}$ is the leading - coefficient of the denominator. Here, $a_{n}=12$ and $b_{m}=4$. So, $y = \frac{12}{4}=3$.

Answer:

A. The horizontal asymptote is $y = 3$