What is the end behavior of f(x)?

How it works

For a polynomial or a rational function f, one term decides the end behavior: f(x) ≈ c xᵏ for large |x|, where c ≠ 0 and k is a whole number (negative for a function that shrinks to 0).

1. f is written as one fraction P/Q (terms over a common bottom; x^−2 is 1/x^2), and a computer algebra system (nerdamer, open source) multiplies out the top P and the bottom Q (each checked numerically at 20 points). Numbers typed with a power of ten go to the algebra as exact powers of ten (1e-20 as 1/10^20), so a tiny coefficient is not lost. The algebra runs in the background after you start typing.
2. The leading term of each is its term with the highest power of x. Its coefficient is the term at x = 1 (π x^2 gives π).
3. k = (degree of P) − (degree of Q), and c = (leading coefficient of P)/(leading coefficient of Q), simplified exactly by the algebra.
4. Check: f(x)/(c xᵏ) is evaluated at x = 10⁴, 10⁵, 10⁶ and at −10⁴, −10⁵, −10⁶. On each side, its distance from 1 at 10⁶ must be at most 10⁻² and no larger than at 10⁴ (plus 10⁻¹²). If not, the leading term has not taken over by 10⁶ and the page says "No verified answer": x² + 10⁻²⁰x³ behaves like 10⁻²⁰x³ only past x = 10²⁰, so the page gives no answer for it.

Then, as x → ∞:

• k > 0: f → ∞ if c > 0, −∞ if c < 0;
• k = 0: f → c (a horizontal asymptote y = c);
• k < 0: f → 0.

As x → −∞, xᵏ has the sign of (−1)ᵏ: for k > 0, f → ∞ if c(−1)ᵏ > 0 and −∞ if it is negative; for k = 0, f → c; for k < 0, f → 0. If the top multiplies out to 0, f is 0 and every answer is 0.

The page shows As x → ∞, f(x) → and As x → −∞, f(x) → (∞, -∞ written with a hyphen, 0, or the exact number c as the algebra writes it) and For large |x|, f(x) behaves like c xᵏ, written exactly.

What you can type

• A number has at most 15 digits in a row. A computer number keeps only about 16 digits, so a longer one (9007199254740993) would stand for a nearby number (9007199254740992), and the page asks for fewer digits instead. Write very large or very small numbers with a power of ten (1e-20).
• A polynomial or a polynomial divided by a polynomial in x: numbers (with decimals), x, + − * /, brackets, and ^ with a whole-number power (x^-2 is allowed). pi and e may appear in coefficients. A number or bracket next to x multiplies: 3x^2 (x − 1).
• Any function (sqrt, ln, sin, …) or a power that is not a whole number gives a message: this page is for polynomials and rational functions.

What gets no answer

• A check that fails: f(x)/(c xᵏ) still far from 1 at x = ±10⁶, as for (x + 10⁷)/x, whose second term is still large there, or x² + 10⁻²⁰x³, whose leading term is still small there.
• A step that finds no formula or takes over 3 seconds.

Worked examples by hand

f(x) = −3x²(x − 1)(x + 4) (OpenStax College Algebra 2e, section 5.2, Example 7). Multiplying out, the leading term is −3x² · x · x = −3x^4: k = 4 is even and c = −3 is negative, so f → −∞ as x → ∞ and as x → −∞.

f(x) = −x⁹ (OpenStax College Algebra 2e, section 5.2, Example 3). The leading term is −x^9: k = 9 is odd and c = −1. As x → ∞, f → −∞; as x → −∞, (−1)⁹ · (−1) = 1 > 0, so f → ∞.

f(x) = (x² + 4x)/(x³ − 8) (OpenStax College Algebra 2e, section 5.6, Example 7(c)). The leading terms are x² and x³, so k = 2 − 3 = −1 and c = 1: f behaves like 1/x and f → 0 at both ends (the horizontal asymptote y = 0).