Problems 105-108 are based on material learned earlier in...

Problems $105-108$ are 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 the domain of $R(x)=\frac{6 x^{2}-11 x-2}{2 x^{2}-x-6}$. Find any horizontal, vertical, or oblique asymptotes.

Problems $105-108$ are 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 the domain of $R(x)=\frac{6 x^{2}-11 x-2}{2 x^{2}-x-6}$. Find any horizontal, vertical, or oblique asymptotes.

Key Concepts

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches positive or negative infinity. They are determined by comparing the degrees of the numerator and the denominator. If the degree of the numerator is less than that of the denominator, the horizontal asymptote is y = 0; if they are equal, the asymptote is the ratio of the leading coefficients.

Vertical Asymptotes

Vertical asymptotes occur at values of x where the function becomes unbounded, typically where the denominator is zero while the numerator is nonzero. Analyzing vertical asymptotes requires solving for the zeros of the denominator and ensuring that these points are not canceled by corresponding factors in the numerator.

Oblique Asymptotes

Oblique, or slant, asymptotes arise when the degree of the numerator is exactly one more than the degree of the denominator. They represent the linear trend of the function at extreme values of x and can be found via polynomial long division of the numerator by the denominator.

Rational Functions

Rational functions are expressions that represent the ratio of two polynomials. They are fundamental in algebra and calculus, particularly when studying their behavior, limits, and continuity. Understanding how the numerator and denominator interact is crucial, especially when analyzing points of discontinuity and asymptotic behavior.

Domain of a Rational Function

The domain of a rational function consists of all real numbers except those that cause the denominator to be zero. Determining the domain involves identifying and excluding these values, as division by zero is undefined in mathematics.

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