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.
$$
h(x)=\frac{x^{2}-3 x-4}{x^{2}-x-6}
$$

Key Concepts

Graphing Rational Functions

Graphing a rational function involves finding its intercepts, asymptotes, and any discontinuities. By combining these elements, one can sketch a rough graph of the function, understanding its behavior near asymptotes and at infinity, as well as across any holes due to factors canceling.

Removable Discontinuities (Holes)

Removable discontinuities, or holes, in the graph of a rational function occur when a factor is common to both the numerator and the denominator. Although the function is undefined at these x-values, the graphical behavior near these points is continuous aside from the missing value.

Slant (Oblique) Asymptotes

A slant, or oblique, asymptote occurs when the degree of the numerator is exactly one more than the degree of the denominator. In this case, polynomial long division is used to find the linear function that the rational function approaches for large absolute values of x.

Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of a rational function as x approaches positive or negative infinity. They are 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; if equal, it is the ratio of the leading coefficients; if greater, no horizontal asymptote exists.

Vertical Asymptotes

Vertical asymptotes occur at the x-values where the denominator of a rational function equals zero after any common factors with the numerator are canceled. These asymptotes represent values that the function cannot take and typically indicate points where the function tends to positive or negative infinity.

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