Written by the Fiveable Content Team • Last updated September 2025
Written by the Fiveable Content Team • Last updated September 2025
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
Vertical asymptotes occur where the denominator of a rational function is zero and the numerator is non-zero.
To find vertical asymptotes, set the denominator equal to zero and solve for $x$.
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.
A rational function can have multiple vertical asymptotes.
Vertical asymptotes are shown as dashed lines on graphs to indicate that the function does not touch or cross these lines.
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.