← Chapter 10
College Algebra · Chapter 10

Asymptotes and Holes: Where the Graph Breaks

A rational function is undefined wherever the bottom is zero — but those spots are not all alike. Some are walls the graph runs along forever; some are a single missing point you would never notice by looking. The factored form tells you which is which, and this walks you through the check one step at a time.

The graph

Zoom out — widen the window
the curve vertical asymptote horizontal asymptote hole slant asymptote
Drag the slider and watch the far ends of the curve settle onto the blue dashed line. That flattening is what a horizontal asymptote means — it is a statement about the ends, not about the middle.

What we have found so far

Domain—
Vertical asymptote—
Horizontal asymptote—
Hole—
Each box fills in as you step through the reasoning on the right.
Pick a functionthen walk the steps

Choose a function

The three tests, in one place

Vertical asymptoteAfter cancelling, a value that makes the bottom zero while the top is not zero. A nonzero number over something shrinking to 0 grows without bound. The graph can never cross it.
HoleA factor that appears on top and bottom and cancels. Substituting gives 0/0. Height of the hole = plug the value into the simplified form.
Horizontal asymptoteCompare degrees. Bottom bigger → y = 0. Degrees equal → y = ratio of leading coefficients. Top bigger → none (one bigger gives a slant line). The graph may cross it.