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$$\lim _{x \rightarrow+\infty}[f(x)-g(x)]=0$$and 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 verticalasymptotes. $$ f(x)=\frac{x^{5}-x^{3}+3}{x^{2}-1} $$

Key Concepts

Asymptotic Curves

The concept of asymptotic curves generalizes the idea of an asymptote beyond a line. Two curves are considered asymptotic if their difference approaches zero as x approaches a particular limit (often infinity). This means that at extreme values of x, the two curves become arbitrarily close to each other, forming a basis for comparing the behavior of complex functions with simpler ones.

Limits at Infinity

Limits at infinity are used to analyze the behavior of functions as x tends to positive or negative infinity. By considering the limit of the difference between two functions, one can determine whether a simpler function adequately describes the asymptotic behavior of a more complicated one. This technique is pivotal in identifying extended asymptotic relationships between curves.

Polynomial Division for Rational Functions

Polynomial division is a method used to simplify rational functions, especially when the degree of the numerator is higher than that of the denominator. By dividing the polynomials, one obtains a quotient that often represents the dominant behavior of the function at infinity. The quotient can serve as the simpler function g(x) to which the original function f(x) is asymptotic.

Vertical Asymptotes

Vertical asymptotes occur in rational functions at values of x where the denominator equals zero and the function grows without bound. Identifying vertical asymptotes involves finding the roots of the denominator and analyzing the behavior of the function near these points to determine whether the function approaches infinity, thereby highlighting important discontinuities in the function's graph.

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