# What is the end behavior of f(x)?

Finds where a polynomial or rational function goes as x → ∞ and x → −∞, and the term it behaves like, checked numerically.

- Page: https://www.acalculator.org/math/end-behavior-calculator
- JSON spec: https://www.acalculator.org/math/end-behavior-calculator.json
- Version: 873db43ba75c

## Default answer

Example with the default inputs (Function f(x) -3x^2 (x - 1)(x + 4)): As x → ∞, -3x^2 (x - 1)(x + 4) → -∞; as x → −∞, it → -∞.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| f | Function f(x) | A polynomial or a polynomial over a polynomial, typed like -3x^2 (x - 1)(x + 4) or (6x^3 - 10x^2)/(x^3 + 5x^2). |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| right | As x → ∞, f(x) → | ∞, −∞, or the number f approaches for large x. |
| left | As x → −∞, f(x) → | ∞, −∞, or the number f approaches for large negative x. |
| like | For large \|x\|, f(x) behaves like | The leading term: c x^k, where f(x)/x^k → c. |

## Method

For a polynomial or rational function, f(x)/x^k → c ≠ 0 as x → ±∞ for one whole number k (the leading term c x^k). A computer algebra system finds c; the ends follow from the signs of c and of x^k.

## Assumptions

- f is a polynomial or a quotient of polynomials in x.
- An answer that fails its check is not shown.

## Worked examples

1. f = -3x^2 (x - 1)(x + 4) gives right = -∞, left = -∞, like = -3x^4. Source: OpenStax, College Algebra 2e, section 5.2 Power Functions and Polynomial Functions, Example 7. https://openstax.org/books/college-algebra-2e/pages/5-2-power-functions-and-polynomial-functions.
2. f = -x^9 gives right = -∞, left = ∞, like = -x^9.
3. f = (x^2 + 4x)/(x^3 - 8) gives right = 0, left = 0, like = 1/x.

## FAQ

### What is end behavior?

The end behavior of a function is what f(x) does as x grows without bound, to the right (x → ∞) and to the left (x → −∞): it may grow to ∞, fall to −∞, or settle at a number, which is then a horizontal asymptote.

### How do I find the end behavior of a polynomial?

Only the leading term matters: for large |x| it outweighs all the others. For −3x²(x − 1)(x + 4) the leading term is −3x⁴. An even power with a negative coefficient falls to −∞ at both ends; an odd power such as −x⁹ goes to −∞ on the right and ∞ on the left.

### What about a rational function?

Divide the leading terms of the top and the bottom. If the top has the higher degree, f behaves like a power of x and grows without bound. If the degrees are equal, f approaches the ratio of the leading coefficients. If the bottom has the higher degree, f approaches 0.

### What does "behaves like" mean?

f(x)/(c xᵏ) → 1 as x → ±∞: the ratio of f to its leading term tends to 1. For (x² + 4x)/(x³ − 8) that term is 1/x, so the function shrinks like 1/x towards 0.

### Why does the page not take sin(x) or eˣ?

Those are not polynomials, and their end behavior does not come from a leading power: eˣ grows faster than any power, and sin(x) keeps oscillating with no limit. The page handles polynomials and quotients of polynomials, where the leading term decides everything.

### How is the answer checked?

The top and the bottom are expanded by a computer algebra system, which checks each expansion numerically. Then f(x)/(c xᵏ) is evaluated at x = ±10⁴, ±10⁵ and ±10⁶ and must approach 1, where c is the leading coefficient. If it does not, the page says "No verified answer".

## Sources

- OpenStax, College Algebra 2e, section 5.2 Power Functions and Polynomial Functions: https://openstax.org/books/college-algebra-2e/pages/5-2-power-functions-and-polynomial-functions
- OpenStax, College Algebra 2e, section 5.6 Rational Functions: https://openstax.org/books/college-algebra-2e/pages/5-6-rational-functions