Question

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

Explanation:

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

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

Step2: Evaluate the limit

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

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

Answer:

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