Find the vertical asymptotes, if any, and the horizontal or oblique asymptote, if any, of y=(x^2

step 1 Okay, so if we have y is equal to x squared minus nine divided by x squared minus four, and we'r

Find the vertical asymptotes, if any, and the horizontal or...

Key Concepts

Rational Functions

A rational function is a quotient of two polynomials. Its analysis involves understanding the behavior both near values that make the denominator zero and at the extremes (as x approaches ±?), where comparisons of degrees of the numerator and denominator are used to determine asymptotic behavior.

Vertical Asymptotes

Vertical asymptotes occur at values of x where the denominator of a rational function is zero, provided that these zeros do not cancel with zeros from the numerator. They represent values at which the function grows without bound or is undefined.

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as x approaches infinity or negative infinity. For rational functions, when the degrees of the numerator and denominator are equal, the horizontal asymptote is found by taking the ratio of their leading coefficients; if the degree of the numerator is less than that of the denominator, the horizontal asymptote is y = 0.

Oblique Asymptotes

Oblique or slant asymptotes arise when the degree of the numerator is exactly one more than the degree of the denominator. In these cases, the long-term behavior of the function resembles a linear function, which can be determined by performing polynomial long division.

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