Write an equation of the slant or nonlinear asymptot $$ f(x)=\frac{-4 x^{3}+8 x^{2}-2 x+5}{x-2} $$ The (Choose one) $$ \nabla $$ asymptote is $$ y=\square $$. slant nonlinear

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Find the horizontal asymptote of the function $$ f(x)=\frac{-4x^3+8x^2-2x+5}{x-2} $$.
Function by following steps:
- step0: Find the horizontal asymptotes:
  $$f\left(xight)=-4x^{3}+8x^{2}-2x+\frac{5}{x-2}$$
- step1: Evaluate the limits $$\lim _{xightarrow +\infty}\left(f\left(xight)ight)$$ and $$\lim _{xightarrow -\infty}\left(f\left(xight)ight):$$
  $$\begin{align}&\lim _{xightarrow +\infty}\left(-4x^{3}+8x^{2}-2x+\frac{5}{x-2}ight)\&\lim _{xightarrow -\infty}\left(-4x^{3}+8x^{2}-2x+\frac{5}{x-2}ight)\end{align}$$
- step2: Calculate:
  $$\begin{align}&-\infty\&+\infty\end{align}$$
- step3: The function has no horizontal asymptotes:
  $$	extrm{No horizontal asymptotes}$$
The given function $$ f(x)=\frac{-4x^3+8x^2-2x+5}{x-2} $$ does not have a horizontal asymptote.

To find the slant or nonlinear asymptote, we need to divide the numerator by the denominator using long division or synthetic division.

Let's perform the long division to find the slant asymptote.
Divide the polynomials by following steps:
- step0: Using Long Division to Divide Polynomials:
  $$\left(-4x^{3}+8x^{2}-2x+5ight)\div \left(x-2ight)$$
- step1: Calculate:
  $$\left(-4x^{2}-2ight)	ext{ }	extrm{remainder}	ext{ }1$$
The result of the long division is $$ -4x^2 - 2 $$ with a remainder of 1.

Therefore, the slant or nonlinear asymptote of the function $$ f(x)=\frac{-4x^3+8x^2-2x+5}{x-2} $$ is $$ y = -4x^2 - 2 $$.
The slant or nonlinear asymptote of the function $$ f(x)=\frac{-4x^3+8x^2-2x+5}{x-2} $$ is $$ y = -4x^2 - 2 $$.

Write an equation of the slant or nonlinear asymptot \[ f(x)=\frac{-4 x^{3}+8 x^{2}-2 x+5}{x-2} \] The (Choose one) \( \nabla \) asymptote is \( y=\square \). slant nonlinear

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