Problems 134–137 are based on material learned earlier in...

Problems 134–137 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 horizontal and vertical asymptotes of $R(x)=\frac{3 x^{2}-12}{x^{2}-5 x-14}$

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Key Concepts

Rational Functions

A rational function is defined by the ratio of two polynomials. Its analysis involves investigating features such as intercepts, domains, and asymptotic behavior, which together help describe the function’s overall characteristics.

Horizontal Asymptote

The horizontal asymptote of a rational function indicates the behavior of the function as the input variable approaches infinity or negative infinity. It is determined by comparing the degrees of the numerator and the denominator, showing the value that the function approaches for large magnitude inputs.

Vertical Asymptote

Vertical asymptotes occur at points where the denominator of a rational function is zero while the numerator remains nonzero, leading the function's value to tend toward infinity. Identifying these helps understand the function's behavior near these points of discontinuity.

Polynomial Factorization

Polynomial factorization involves breaking down a polynomial into a product of its simpler factors, often linear or quadratic. This technique is useful for simplifying expressions, solving equations, and determining the roots or zeros of a polynomial, which in turn helps in detecting potential asymptotes in rational functions.

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