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 $x$ 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 ($y = 0$) 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 $f(x) = \frac{a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0}{b_m x^m + b_{m-1} x^{m-1} + \cdots + b_0}$, compare $n$ and $m$.

Leading Coefficients: If $n = m$, the horizontal asymptote is $y = \frac{a_n}{b_m}$.

No Horizontal Asymptote: If $n > m$, there is no horizontal asymptote.

Problem: Find the horizontal asymptote of $f(x) = \frac{2x + 3}{4x - 5}$.

The degrees of both the numerator and the denominator are 1.
The horizontal asymptote is given by the ratio of the leading coefficients: $y = \frac{2}{4} = \frac{1}{2}$.

Validation: The function approaches $y = \frac{1}{2}$ as $x$ approaches infinity.

Problem: Determine the horizontal asymptote of $g(x) = \frac{3x^2 + 2x - 1}{x^2 - 4}$.

The degrees of both the numerator and the denominator are 2.
The horizontal asymptote is given by the ratio of the leading coefficients: $y = \frac{3}{1} = 3$.

Validation: The function approaches $y = 3$ as $x$ 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 $x$ to confirm the horizontal asymptote.