# [[Vertical Asymptote]] (Mathematics > Algebra) ## Definition A vertical asymptote is a vertical line $x = a$ where the function $f(x)$ increases or decreases without bound as $x$ approaches $a$. The function gets arbitrarily close to the vertical asymptote but never touches or crosses it. Vertical asymptotes occur where the denominator of a rational function equals zero and the numerator does not also equal zero. For example, for the function: $ f(x) = \frac{1}{(x - 2)(x + 2)} $ Vertical asymptotes occur at $x = 2$ and $x = -2$. ## Key Concepts - **Approaching Infinity**: As $x$ approaches the asymptote, the function's value tends to $\infty$ or $-\infty$. - **Denominator Zero**: Vertical asymptotes occur at values of $x$ where the denominator is zero. - **No Crossing**: The function does not cross a vertical asymptote, though it may come arbitrarily close. ## Important Properties 1. The function tends toward infinity or negative infinity as it approaches the asymptote. 2. The asymptote represents a boundary for the function, but the function does not cross it. 3. Vertical asymptotes are found by setting the denominator of a rational function equal to zero and solving for $x$ (if the numerator is non-zero). ## Essential Formulas - Given a rational function $f(x) = \frac{p(x)}{q(x)}$, vertical asymptotes occur where: $ q(x) = 0 \quad \text{and} \quad p(x) \neq 0 $ ## Core Examples 1. **Simple Example**: Consider the function $f(x) = \frac{1}{x - 3}$. Set the denominator to zero: $ x - 3 = 0 \quad \Rightarrow \quad x = 3 $ So, there is a vertical asymptote at $x = 3$. 2. **Advanced Example**: For the function $f(x) = \frac{x^2}{x^2 - 4}$, factor the denominator: $ f(x) = \frac{x^2}{(x - 2)(x + 2)} $ Vertical asymptotes occur at $x = 2$ and $x = -2$. ## Related Theorems/Rules - **Limits at Infinity**: As $x$ approaches a vertical asymptote, $\lim_{x \to a^{+}} f(x) = \infty$ or $\lim_{x \to a^{-}} f(x) = -\infty$ depending on the function's behavior. ## Common Pitfalls - Forgetting that vertical asymptotes occur only if the numerator is non-zero at that point. - Misidentifying removable discontinuities as vertical asymptotes. ## Related Topics - [[Horizontal Asymptote]] - [[Rational Functions]] ## Quick Review Questions 1. How do you find the vertical asymptotes of a rational function? 2. What happens to the value of the function as $x$ approaches a vertical asymptote? *** ![[Vertical_Asymptote_visualization]]