Asymptotes Find all horizontal and vertical asymptotes (if...

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Key Concepts

Rational Functions

Rational functions are expressions formed by dividing one polynomial by another. The behavior of these functions, such as asymptotes and discontinuities, is largely determined by the degrees and factors of the numerator and denominator polynomials.

Vertical Asymptotes

Vertical asymptotes occur at the values of the variable that make the denominator zero, provided these zeros are not canceled out by similar factors in the numerator. They represent values where the function tends to ±infinity and indicate points of discontinuity in the graph.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as the variable approaches infinity or negative infinity. They are determined by comparing the degrees of the numerator and the denominator. In cases where the degree of the numerator is less than or equal to that of the denominator, special rules apply to determine the horizontal asymptote.

Degree of Polynomials and Asymptotic Behavior

The degrees of the numerator and denominator in a rational function are crucial in determining the asymptotic behavior. When the numerator’s degree is higher by exactly one, the function typically has an oblique (slant) asymptote instead of a horizontal one. This concept helps in understanding the end behavior of the function.

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