Question

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

Explanation:

Step1: Divide numerator and denominator by \(x^3\)

$$\begin{align*} \lim_{x ightarrow\pm\infty}f(x)&=\lim_{x ightarrow\pm\infty}\frac{\frac{5x}{x^3}-\frac{x^3}{x^3}-\frac{5}{x^3}}{\frac{-4x^3}{x^3}+\frac{3x^2}{x^3}-\frac{3}{x^3}}\\ &=\lim_{x ightarrow\pm\infty}\frac{\frac{5}{x^2}-1-\frac{5}{x^3}}{-4 +\frac{3}{x}-\frac{3}{x^3}} \end{align*}$$

Step2: Evaluate the limit

As \(x
ightarrow\pm\infty\), \(\frac{5}{x^2}
ightarrow0\), \(\frac{5}{x^3}
ightarrow0\), \(\frac{3}{x}
ightarrow0\), \(\frac{3}{x^3}
ightarrow0\)

$$ \lim_{x ightarrow\pm\infty}\frac{\frac{5}{x^2}-1-\frac{5}{x^3}}{-4 +\frac{3}{x}-\frac{3}{x^3}}=\frac{0 - 1-0}{-4+0 - 0}=\frac{-1}{-4}=\frac{1}{4} $$

Answer:

\(y = \frac{1}{4}\)