Horizontal Asymptotes - Algebra-2

1. Fundamental Concepts

Definition: A horizontal asymptote is a horizontal line that the graph of a function approaches as the input (x) increases or decreases without bound.
Degree Comparison: For rational functions, the behavior of the function as approaches infinity depends on the degrees of the numerator and denominator.
Horizontal Asymptote Rules:
- If the degree of the numerator is less than the degree of the denominator, the x-axis () is the horizontal asymptote.
- If the degrees are equal, the horizontal asymptote is given by the ratio of the leading coefficients.
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.

Degree Analysis: For , compare and .

Leading Coefficients: If , the horizontal asymptote is .

No Horizontal Asymptote: If , there is no horizontal asymptote.

Problem: Find the horizontal asymptote of .

Validation: The function approaches as approaches infinity.

Problem: Determine the horizontal asymptote of .

Validation: The function approaches as approaches infinity.

Compare Degrees: Always start by comparing the degrees of the numerator and the denominator to determine the type of asymptote.
Leading Coefficient Rule: Use the leading coefficient rule when the degrees are equal.
Graphical Interpretation: Visualize the function and its behavior at extreme values of to confirm the horizontal asymptote.