Problems 93-96 are based on material learned earlier in the...

Problems 93-96 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 any asymptotes for the graph of without graphing. $$ R(x)=\frac{x+3}{x^2-x-12} \text {. } $$

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Key Concepts

Removable Discontinuities

Removable discontinuities, also known as holes, occur when a factor in the numerator cancels with a corresponding factor in the denominator. Although the simplified function might appear continuous, the original function is not defined at that specific x-value, creating a 'hole' in the graph.

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; for instance, if the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.

Rational Functions

Rational functions are expressions formed by dividing one polynomial by another. Their overall behavior, including continuity and asymptotic tendencies, is influenced by the degrees of the polynomials in the numerator and denominator as well as by any common factors that may simplify the expression.

Vertical Asymptotes

Vertical asymptotes are lines x = a where a rational function’s denominator equals zero and the corresponding factor does not cancel out with the numerator. They represent values of x at which the function grows without bound, indicating points of discontinuity in the graph.

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