Vertical asymptote

Definition

A vertical asymptote is a line $x = a$ where a rational function $f(x)$ approaches positive or negative infinity as $x$ approaches $a$. It represents values that $x$ cannot take, causing the function to become unbounded.

5 Must Know Facts For Your Next Test

1. Vertical asymptotes occur where the denominator of a rational function is zero and the numerator is non-zero.
2. To find vertical asymptotes, set the denominator equal to zero and solve for $x$.
3. If both the numerator and denominator have common factors that can be canceled out, those points are not vertical asymptotes but holes in the graph.
4. A rational function can have multiple vertical asymptotes.
5. Vertical asymptotes are shown as dashed lines on graphs to indicate that the function does not touch or cross these lines.

Review Questions

• How do you determine if a rational function has a vertical asymptote at a particular value of $x$?
• What happens to the value of a rational function as it approaches its vertical asymptote?
• Can a rational function have more than one vertical asymptote? Explain with an example.

Related terms

Horizontal Asymptote: A horizontal line $y = b$ where a rational function approaches as $x$ tends towards positive or negative infinity.

Hole (in Graph): A point on the graph of a rational function where both numerator and denominator are zero after canceling common factors, resulting in undefined value at that point.

Oblique Asymptote: A slanted line that a rational function approaches but never touches or crosses as x goes to positive or negative infinity. Occurs when degree of numerator is one more than degree of denominator.