When does an oblique asymptote occur in relation to horizontal asymptotes?

An oblique asymptote occurs when there is no horizontal asymptote, specifically in the context of rational functions. This situation typically arises when the degree of the numerator is greater than the degree of the denominator by exactly one. In such cases, as the variable approaches infinity, the function behaves like a linear polynomial rather than approaching a specific horizontal line.

Horizontal asymptotes describe the behavior of a function as it approaches a particular y-value, regardless of how large or small the input becomes. For a rational function to have a horizontal asymptote, the degree of the numerator must be less than or equal to the degree of the denominator. When the degree of the numerator exceeds the degree of the denominator, it effectively means the function will not settle on a single horizontal value and will instead have an oblique (or slant) asymptote. Thus, the presence of an oblique asymptote directly indicates that a horizontal asymptote does not exist in that scenario.