Graphs of Rational Functions

For each functon find the following (if they exist).

     a. End Behavior including the equations of horizontal or slant asymptotes.

     b. Vertical Asymptote(s). Distinguish between VA and Holes.

\(y=\dfrac{-3x^3}{x^3-4x}\\ y=\dfrac{-3x^3}{x(x-2)(x+2)}\\~\\ so\;\;x\ne\pm2,\;\;\;x\ne0\\~\\ y=\dfrac{-3x^3}{x^3-4x}\\ \)

\(y=\dfrac{-3x^3\div x^3}{(x^3-4x)\div x^3}\\ y=\dfrac{-3}{1-\frac{4}{x^2}}\\ \displaystyle\lim_{x\rightarrow 0^\pm}y=\frac{-3}{1-\infty}=0 \quad \text{Hole at (0,0)} \\\displaystyle\lim_{x\rightarrow \pm\infty}y=\frac{-3}{1-0}=-3\\~\\ \text{Let }\delta\;\;\text{ be a miniscule positive number.}\\ \displaystyle\lim_{x\rightarrow2^+}y=\frac{-3}{1-(1-\delta)}=\frac{-3}{\delta}=-\infty\\ \displaystyle\lim_{x\rightarrow-2^-}y=\frac{-3}{1-(1-\delta)}=\frac{-3}{\delta}=-\infty\\ \displaystyle\lim_{x\rightarrow2^-}y=\frac{-3}{1-(1+\delta)}=\frac{-3}{-\delta}=\infty\\ \displaystyle\lim_{x\rightarrow-2^+}y=\frac{-3}{1-(1+\delta)}=\frac{-3}{-\delta}=\infty\\ \)
 

Hole at (0,0)

Vertical asymptotes at  x=+2 and x=-2

Horizontal asymptotes at  y=-3

Here is the graph