Question

write the equation of the horizontal asymptote of the function: $y = \frac{2x^{2}-3x - 5}{4x^{2}-1}$

Explanation:

Step1: Determine the degrees of polynomials

The degree of the numerator $2x^{2}-3x - 5$ is $n = 2$, and the degree of the denominator $4x^{2}-1$ is $m=2$.

Step2: Use the 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, $a_{n}=2$ and $b_{m}=4$.
So, $y=\frac{2}{4}=\frac{1}{2}$.

Answer:

$y = \frac{1}{2}$