Based on material learned earlier in the course. The purpose...

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Find any asymptotes for the graph of
$$
R(x)=\frac{x+3}{x^{2}-x-12}
$$

Key Concepts

Rational Functions

Rational functions are ratios of two polynomials. They are central to algebra and precalculus, where understanding the behavior of these functions—including their intercepts, discontinuities, and end behavior—is crucial. Analysis of rational functions involves investigating their domains, simplifying expressions, and graphing their characteristics.

Vertical Asymptotes

Vertical asymptotes occur for rational functions where the function approaches infinity or negative infinity as the variable approaches a specific value, typically where the denominator is zero and the numerator is nonzero (after any common factors have been canceled). They are indicators of values that are not in the domain of the function, and they represent lines that the graph approaches but never crosses.

Horizontal Asymptotes

Horizontal asymptotes describe the end behavior of a rational function as the variable approaches positive or negative infinity. Determining the horizontal asymptote involves comparing the degrees of the numerator and the denominator. This concept helps to understand how the function levels off or approaches a constant value at extreme values of the variable.

Oblique Asymptotes

Oblique (or slant) asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator in a rational function. In such cases, polynomial long division is used to find the linear function that the graph of the rational function approaches as the variable goes to infinity. These asymptotes describe a non-horizontal, non-vertical end behavior.

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