How-to · Updated October 2, 2026
What is the end behavior of a rational function?
Direct answer
If the denominator degree is higher, both ends approach 0. If the degrees match, both ends approach the ratio of the leading coefficients. If the numerator is higher, the ends follow that quotient.
Three degree cases
| Example | Degrees | As x → −∞ | As x → +∞ |
|---|---|---|---|
| 1/x | 0 over 1 | 0 | 0 |
| 1/x² | 0 over 2 | 0 | 0 |
| 2x / x | 1 over 1 | 2 | 2 |
| x² / x | 2 over 1 | -∞ | +∞ |
| −2x³ / x | 3 over 1 | -∞ | -∞ |
As x → ±∞, f(x) → 0. As x → ±∞, f(x) → 2. −2x³ / x matches a degree-2 polynomial with leading coefficient −2, so As x → -∞, f(x) → -∞ and as x → +∞, f(x) → -∞.
Limits
- End behavior is the limit at infinity. Vertical asymptotes and holes are separate from these ends.
- 1/x approaches 0 from below on the left and from above on the right. The horizontal asymptote is still y = 0.
- A slant asymptote describes the line the graph follows. The function values still grow without a finite limit.
Use the calculator
For a polynomial with no denominator: the leading coefficient test. Enter 1/x or 1/x² in the end behavior calculator.