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})

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

Step2: Find the limit as (x\to\pm\infty)

As (x\to\pm\infty), (\frac{5}{x^{2}}\to0), (\frac{5}{x^{3}}\to0), (\frac{3}{x}\to0), (\frac{3}{x^{3}}\to0) $$ \lim_{x\to\pm\infty}f(x)=\frac{0 - 1-0}{-4+0 - 0}=\frac{-1}{-4}=\frac{1}{4} $$

Answer:

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