The notion of an asymptote can be extended to include curves...

The notion of an asymptote can be extended to include curves as well as lines. Specifically, we say that curves $y=f(x)$ and $y=g(x)$ are asymptotic as $x \rightarrow+\infty$ provided $$ \lim _{x \rightarrow+\infty}[f(x)-g(x)]=0 $$ In these exercises, determine a simpler function $g(x)$ such that $y=f(x)$ is asymptotic to $y=g(x)$ as $x \rightarrow+\infty$ or $x \rightarrow-\infty$ Use a graphing utility to generate the graphs of $y=f(x)$ and $y=g(x)$ and identify all vertical asymptotes. $$ f(x)=\frac{-x^{3}+3 x^{2}+x-1}{x-3} $$

The notion of an asymptote can be extended to include curves as well as lines. Specifically, we say that curves $y=f(x)$ and $y=g(x)$ are asymptotic as $x \rightarrow+\infty$ provided
$$
\lim _{x \rightarrow+\infty}[f(x)-g(x)]=0
$$
In these exercises, determine a simpler function $g(x)$ such that $y=f(x)$ is asymptotic to $y=g(x)$ as $x \rightarrow+\infty$ or $x \rightarrow-\infty$ Use a graphing utility to generate the graphs of $y=f(x)$ and $y=g(x)$ and identify all vertical asymptotes.
$$
f(x)=\frac{-x^{3}+3 x^{2}+x-1}{x-3}
$$

Key Concepts

Asymptotic Behavior

This concept involves understanding how functions behave as the input variable approaches infinity, negative infinity, or other critical points. In the context of the problem, two functions are said to be asymptotic if their difference approaches zero as x tends to infinity (or negative infinity), which allows for approximating complicated functions with simpler ones over long intervals.

Curve Asymptotes

Unlike traditional asymptotes which are straight lines, curve asymptotes involve approximating a function by another function (often simpler or of lower degree) over a limit process. This idea extends asymptotic analysis to include scenarios where the neighboring graph of a function behaves similarly to a non-linear curve as x approaches a boundary value.

Polynomial Division

This algebraic technique is used to simplify rational functions by dividing a higher degree polynomial by a lower degree polynomial. In problems like the one given, performing polynomial division helps identify the dominant term or simpler function that approximates the original function's behavior as x becomes very large.

Vertical Asymptotes

Vertical asymptotes occur where a function becomes unbounded as the input approaches specific values, typically when there is a zero in the denominator that is not canceled by a similar factor in the numerator. The analysis of vertical asymptotes is crucial in understanding the complete behavior of rational functions across their domain.

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