Question

find the horizontal asymptote of $f(x)=\frac{x - 4x^{3}-5}{-x^{3}+2x^{2}+3}$.
$y =$
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Explanation:

Step1: Identify degrees of polynomials

The degree of the numerator $n = 3$ (highest - power of $x$ is $3$ in $-4x^{3}$) and the degree of the denominator $m = 3$ (highest - power of $x$ is $3$ in $-x^{3}$).

Step2: Find the horizontal asymptote formula

When $n = m$, the horizontal asymptote $y$ is given by the ratio of the leading - coefficients of the numerator and the denominator. The leading - coefficient of the numerator $a=-4$ and the leading - coefficient of the denominator $b = - 1$.

Step3: Calculate the ratio

$y=\frac{a}{b}=\frac{-4}{-1}=4$.

Answer:

$y = 4$