In Exercises 71–78, find the vertical asymptotes, if any,...

In Exercises 71–78, find the vertical asymptotes, if any, the horizontal asymptote, if one exists, and the slant asymptote, if there is one, of the graph of each rational function. Then graph the rational function.
$$
f(x)=\frac{2 x}{x^{2}-9}
$$

Key Concepts

Graphing Rational Functions

Graphing rational functions involves identifying key features such as vertical, horizontal, and slant asymptotes, as well as intercepts and potential holes. This process helps in visualizing the function's behavior, particularly its behavior near discontinuities and for large absolute values of x.

Slant Asymptotes

Slant (or oblique) asymptotes occur when the degree of the numerator is exactly one higher than the degree of the denominator. They are found using polynomial long division, and they describe the asymptotic behavior of the function in cases where the function does not have a horizontal asymptote.

Horizontal Asymptotes

A horizontal asymptote describes the behavior of a rational function as x tends toward positive or negative infinity. It is determined by comparing the degrees of the numerator and denominator. If the degree of the numerator is less than that of the denominator, the horizontal asymptote is y = 0, while if the degrees are equal, it is the ratio of the leading coefficients.

Vertical Asymptotes

Vertical asymptotes occur at the x-values where the denominator of a rational function is zero (and the corresponding factors are not canceled by the numerator). They represent values where the function grows without bound or drops to negative infinity. These asymptotes help in understanding the limits and discontinuities of a function.

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