Rational functions and asymptotes

A rational function is one polynomial divided by another, like f(x) = 1 / (x − 1). Dividing introduces something new: lines the graph rushes toward but never touches — asymptotes.

Vertical asymptotes: where the bottom hits zero

A fraction blows up when its denominator approaches zero. So wherever the bottom equals zero (and the top doesn't), the function shoots off to ±∞:

For f(x) = 1 / (x − 1), the denominator is zero at x = 1. That input is banned from the domain, and the graph hugs the vertical line x = 1 without ever crossing it.

Horizontal asymptotes: where the graph heads far out

Push x toward ±∞ and ask what height the graph settles at. Compare the degrees of top and bottom:

For 1 / (x − 1), the top (degree 0) is smaller than the bottom (degree 1), so the horizontal asymptote is y = 0 — far out in either direction, the curve flattens toward the x-axis.

A subtlety: holes vs. asymptotes

If a factor cancels from top and bottom, you get a hole, not an asymptote. For (x − 2)/((x − 2)(x + 1)), the (x − 2) cancels: there's a hole at x = 2 and a true vertical asymptote only at x = −1. Always factor first.

Why this matters

Asymptotes describe limits in disguise — "approaches but never reaches" is the language of the final module. They also model the real world: a drug's concentration decaying toward zero, or a population leveling off at the carrying capacity its environment can support.