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# Graphs of Rational Functions

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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.

$$f(x)=\frac{-3x^3}{x^3-4x}$$

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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

Melody  Nov 15, 2017

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