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

Explanation:

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

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

Step2: Evaluate the limit

As (x\rightarrow\pm\infty), (\frac{5}{x^2}\rightarrow0), (\frac{5}{x^3}\rightarrow0), (\frac{3}{x}\rightarrow0), (\frac{3}{x^3}\rightarrow0) $$ \lim_{x\rightarrow\pm\infty}\frac{\frac{5}{x^2}-1-\frac{5}{x^3}}{-4 +\frac{3}{x}-\frac{3}{x^3}}=\frac{0 - 1-0}{-4+0 - 0}=\frac{-1}{-4}=\frac{1}{4} $$

Answer:

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