End behavior and infinity notation
End Behavior and Infinity Notation
End Behavior and Infinity Notation is a topic in Functions in the Texas Essential Knowledge and Skills (Precalculus, §111.42). It is aligned to Standard P.2(J), which requires students to analyze the end behavior of functions using infinity notation.
End behavior describes what a function does as \(x\to\pm\infty\), written with infinity notation; for a polynomial it is set by the leading term, and for a rational function by the degree comparison.
Theory
For a polynomial, only the leading term matters at the extremes. Its degree (even/odd) and the sign of its coefficient fix both ends:
- Even degree: both ends go the same way (up if the coefficient is positive, down if negative).
- Odd degree: the ends go opposite ways.
For a rational function, compare the degrees of numerator and denominator to find the horizontal asymptote that governs end behavior.
Horizontal asymptote of a rational function \(\dfrac{p(x)}{q(x)}\) by degree:
where \(a,b\) are the leading coefficients of \(p,q\).
How to determine end behavior
- Polynomial: find the leading term; use its degree (even/odd) and sign to state both ends.
- Rational: compare \(\deg p\) and \(\deg q\) using the rule above.
- Write it with infinity notation for each direction \(x\to+\infty\) and \(x\to-\infty\).
End behavior is set by the leading term \(-2x^3\) (odd degree, negative coefficient).
| \(x\to+\infty\) | \(\Rightarrow\) | \(f(x)\to-\infty\) |
| \(x\to-\infty\) | \(\Rightarrow\) | \(f(x)\to+\infty\) |
The graph falls on the right and rises on the left.
The leading term \(3x^4\) has even degree and positive coefficient, so both ends go the same way — upward.
| \(x\to\pm\infty\) | \(\Rightarrow\) | \(f(x)\to+\infty\) |
When numerator and denominator have the same degree, the horizontal asymptote is the ratio of leading coefficients.
| \(y\) | \(=\) | \(\dfrac{2}{1}=2\) |
So \(f(x)\to 2\) as \(x\to\pm\infty\).
The numerator degree (1) is less than the denominator degree (2), so the function tends to 0 at the ends.
| \(x\to\pm\infty\) | \(\Rightarrow\) | \(f(x)\to 0\) |
The horizontal asymptote is \(y=0\).
Common pitfalls
Frequently asked questions
What is end behavior?
How a function behaves as \(x\to+\infty\) and \(x\to-\infty\) — whether the outputs rise, fall, or level off at the far left and right.
How do you find the end behavior of a polynomial?
Look only at the leading term. Its degree (even or odd) and the sign of its coefficient determine what both ends do.
How do you find the horizontal asymptote of a rational function?
Compare degrees: smaller top gives \(y=0\); equal degrees give the ratio of leading coefficients; larger top gives no horizontal asymptote.
What does f(x) tends to infinity mean?
The output grows without bound as \(x\) moves toward some value or toward infinity. Infinity is a direction of unbounded growth, not a reachable number.