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$f\left(x\right)\=\frac{x^3\+{\left|x\right|}^3-6x^2\+12}{{\left(x\+2\right)}^2}$$\stackrel{\text{assign as function}}{\to }$$f$

Graph $f\left(x\right)$ 

$g\left(x\right)\&equals;\&lcub;\begin{array}{cc}-\frac{6\left(x^2-2\right)}{{\left(x\&plus;2\right)}^2}& x<0\\ \frac{2\left(x^3-3x^2\&plus;6\right)}{{\left(x\&plus;2\right)}^2}& x\&gt;0\end{array}$

Obtain the rules in $g\left(x\right)$ 

$\mathrm{simplify}\left(f\left(x\right)\right) assuming x<0$

$-\frac{6\left(x^2-2\right)}{{\left(x\&plus;2\right)}^2}$

$h≔\mathrm{simplify}\left(f\left(x\right)\right) assuming x\&gt;0$

$\frac{2\left(x^3-3x^2\&plus;6\right)}{{\left(x\&plus;2\right)}^2}$

Obtain the asymptotes

$\mathrm{Asymptotes}\left(f\left(x\right)\&comma;x\right)$

$\left[y\&equals;2x-14\&comma;y\&equals;-6\&comma;x\&equals;-2\right]$

Obtain the horizontal asymptote $y\&equals;-6$ 

$\underset{x\to -\infty }{lim}f\left(x\right)$ = $-6$$$

Obtain the vertical asymptote $x\&equals;-2$ 

$\underset{x\to -2}{lim}f\left(x\right)$ = $-\mathrm{\&infin;}$

Obtain the oblique asymptote $y\&equals;2 x-14$ 

$\mathrm{quo}\left(\mathrm{numer}\left(h\right)\&comma;\mathrm{denom}\left(h\right)\&comma;x\&comma;\&apos;r\&apos;\right)$ = $2x-14$$$

$2 x-14\&plus;\frac{r}{{\left(x\&plus;2\right)}^2}$ = $2x-14\&plus;\frac{68\&plus;48x}{{\left(x\&plus;2\right)}^2}$$\stackrel{\text{simplify}}{\&equals;}$$\frac{2\left(x^3-3x^2\&plus;6\right)}{{\left(x\&plus;2\right)}^2}$ = $h$$$