Far from the origin, only the leading terms matter. The two degrees decide everything.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
Denominator degree larger
The bottom outgrows the top, so the outputs shrink to zero. The asymptote is y = 0.
Degrees equal
The asymptote is the ratio of the leading coefficients — not zero, and not one unless they match.
Numerator degree larger
There is no horizontal asymptote. If it is larger by exactly one, there is a slant asymptote instead.
Compare the degrees
Numerator degree lower gives a horizontal asymptote at zero. Equal gives the ratio of the leading coefficients. Higher gives no horizontal asymptote — the function grows without bound.
Why the degrees decide
Divide numerator and denominator by the highest power in the denominator. Every term with x underneath tends to zero, leaving only the leading terms to determine the outcome. The rule is that division, done once.
A graph may cross a horizontal asymptote
Horizontal asymptotes describe long-run behaviour only. A rational function can cross one in the middle of its domain and still approach it at the ends. Vertical asymptotes, by contrast, can never be crossed.
Say it in limit language
"As x increases without bound, f approaches 3" is the phrasing expected. Learning to state end behaviour precisely now makes the transition to limit notation in calculus a change of symbols rather than of ideas.
Step 2: Try It Yourself
Tap and try it out.
Step 3: Watch an Example
One step at a time.
Watch Priya Compare Degrees
Priya finds the horizontal asymptote of (3x² + 1) / (2x² − 5).
- Step 1
The numerator has degree 2 and so does the denominator.
Step 4: Your Turn
Practice makes it stick.
Bottom Heavy
Problem 1 of 2
For x / (x² + 1), what is the horizontal asymptote? Give the y value.
Equal Degrees
Problem 2 of 2
For (6x + 1) / (3x − 2), what is the horizontal asymptote? Give the y value.
Compare the Degrees
1 of 8
5/(x + 2). Horizontal asymptote y = ?
2 of 8
(4x² − 1)/(x² + 3). Horizontal asymptote y = ?
3 of 8
(x³ + 1)/(x + 2). How many horizontal asymptotes?
4 of 8
Sort each function by its end behaviour.
Tap something to move it.
- Empty
- Empty
- Empty
5 of 8
(2x + 7)/(5x − 1). Horizontal asymptote y = ? Give it as a decimal.
6 of 8
Top degree exceeds bottom by exactly 1. Is there a slant asymptote? 1 for yes, 0 for no.
7 of 8
(x² + 5)/(x² + 5). Horizontal asymptote y = ?
8 of 8
Can a graph cross its horizontal asymptote? 1 for yes, 0 for no.
Step 5: Quick Check
Show what you know.
Question 1 of 1
(9x² + 2)/(3x² − 7). Horizontal asymptote y = ?
What You Learned
- End behaviour of a rational function is decided by comparing the two degrees.
- Bottom heavier gives y = 0; equal degrees give the ratio of leading coefficients.
- Top heavier gives no horizontal asymptote, and a slant one when it exceeds by exactly 1.