# [[Horizontal Asymptote]] (Mathematics > Calculus) ## Definition A **horizontal asymptote** is a horizontal line that a function approaches as the input (usually $x$) tends to infinity ($x \to \infty$) or negative infinity ($x \to -\infty$), but never touches or crosses (in most cases). Formally, the line $y = L$ is a horizontal asymptote of a function $f(x)$ if: $ \lim_{x \to \infty} f(x) = L \quad \text{or} \quad \lim_{x \to -\infty} f(x) = L $ ## Key Concepts - Horizontal asymptotes describe the end behavior of a function as $x \to \infty$ or $x \to -\infty$. - A function can have at most two horizontal asymptotes (one as $x \to \infty$, one as $x \to -\infty$). - Rational functions often exhibit horizontal asymptotes, determined by the degrees of the polynomials in the numerator and denominator. ## Important Properties 1. **Degree Comparison**: For rational functions, the horizontal asymptote is determined by the degrees of the numerator and denominator polynomials. - If the degree of the numerator is less than the degree of the denominator, the asymptote is $y = 0$. - If the degrees are equal, the asymptote is $y = \frac{a}{b}$, where $a$ and $b$ are the leading coefficients of the numerator and denominator. - If the degree of the numerator is greater, no horizontal asymptote exists. 2. Horizontal asymptotes can occur in one or both directions (as $x \to \infty$ and $x \to -\infty$). 3. A function can approach a horizontal asymptote from above or below. ## Essential Formulas - For the rational function $f(x) = \frac{2x^2}{x^2 - 1}$, the horizontal asymptote is: $ \lim_{x \to \infty} f(x) = 2 $ - General formula for horizontal asymptote of a rational function $\frac{P(x)}{Q(x)}$: $ y = \frac{\text{leading coefficient of } P(x)}{\text{leading coefficient of } Q(x)} $ ## Core Examples 1. **Basic Example**: For the rational function $f(x) = \frac{1}{x}$, as $x \to \infty$ or $x \to -\infty$, $f(x) \to 0$. Therefore, the horizontal asymptote is $y = 0$. 2. **Advanced Example**: For $f(x) = \frac{3x^2}{2x^2 + 1}$, as $x \to \infty$, the horizontal asymptote is: $ \lim_{x \to \infty} f(x) = \frac{3}{2} $ Thus, the horizontal asymptote is $y = \frac{3}{2}$. ## Related Theorems/Rules - **Limits at Infinity**: Horizontal asymptotes are linked to limits of functions as $x \to \infty$ or $x \to -\infty$. If $\lim_{x \to \infty} f(x) = L$, then $y = L$ is a horizontal asymptote. - **End Behavior of Polynomials**: Polynomials themselves do not have horizontal asymptotes, but rational functions do based on the comparison of their polynomial degrees. ## Common Pitfalls - Confusing **vertical asymptotes** (where the function becomes infinite) with horizontal asymptotes (where the function approaches a finite value). - Assuming that functions can cross their horizontal asymptote. While this is true for certain functions, most rational functions do not cross their horizontal asymptote. ## Related Topics - [[Vertical Asymptote]] - [[End Behavior of Functions]] ## Quick Review Questions 1. How do you determine the horizontal asymptote of a rational function where the degree of the numerator and denominator are equal? 2. What is the horizontal asymptote of $f(x) = \frac{5x}{x^2 + 3}$ as $x \to \infty$? *** ![[Horizontal_Asymptote_visualization]]