What is an oblique asymptote?

An oblique asymptote, also known as a slant asymptote, is a type of asymptotic behavior that occurs when the graph of a function approaches a straight line as x approaches positive or negative infinity. Unlike horizontal or vertical asymptotes, oblique asymptotes are not limited to being horizontal or vertical lines.

To determine if a function has an oblique asymptote, one can look for a quotient of two polynomials where the degree of the polynomial in the numerator is exactly one more than the degree of the polynomial in the denominator. In this case, the function will have an oblique asymptote.

The equation of an oblique asymptote can be found by performing polynomial division on the given function. The quotient obtained from the division represents the equation of the oblique asymptote.

Oblique asymptotes are common in rational functions where the degree of the numerator is greater than the degree of the denominator by one. They help describe the long-term behavior of the function as x approaches infinity.