Find all horizontal and vertical asymptotes for each...

Key Concepts

Rational Functions

A rational function is defined as the ratio of two polynomials. Its general behavior, including discontinuities and asymptotes, is largely influenced by the properties of these polynomials. Understanding the structure and factorization of both the numerator and denominator is crucial in determining the function’s behavior.

Vertical Asymptotes

Vertical asymptotes occur at values of the variable where the denominator of a rational function becomes zero, provided that the numerator does not also become zero at those points. This leads to the function's value approaching positive or negative infinity near those specific input values, reflecting potential discontinuities in the function.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a rational function as the input variable approaches infinity or negative infinity. They indicate the value that the function approaches in the long term. The determination of a horizontal asymptote typically depends on the relationship between the degrees of the numerator and the denominator.

Degree Comparison in Rational Functions

Comparing the degrees of the numerator and the denominator is a fundamental method used to determine the existence and nature of horizontal or slant asymptotes. When the degrees are equal, the horizontal asymptote is found by taking the ratio of the leading coefficients. If the numerator’s degree is less than the denominator's, the horizontal asymptote is y=0, while a higher numerator degree usually implies a slant asymptote or other behavior.

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