Introduction
A polynomial’s values can become very large in magnitude as the input moves far to the right or far to the left. To describe this end behavior, we ask what happens to the polynomial as \(x\to+\infty\) and as \(x\to-\infty\).
For a polynomial of positive degree, the answer is determined by two features: the degree’s parity (whether it is even or odd) and the sign of the leading coefficient. Lower-degree terms can affect values for particular inputs, but they do not change the polynomial’s eventual direction at either end.
Degree and Leading Coefficient
A nonzero polynomial can be written in standard form as \(P(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0\), where the coefficients are real numbers, \(n\) is a nonnegative integer, and \(a_n\ne 0\). Its degree is \(n\), and \(a_n\) is its leading coefficient. The leading term is \(a_nx^n\).
If the coefficient of a stated highest power is zero, a lower nonzero power determines the degree. For example, \(0x^5+2x^2-3\) has degree \(2\), not degree \(5\). The zero polynomial, whose coefficients are all zero, has no degree under the usual convention.
The notation \(x\to+\infty\) means that \(x\) grows without bound in the positive direction; \(x\to-\infty\) means that \(x\) decreases without bound. Saying \(P(x)\to+\infty\) means that \(P(x)\) eventually exceeds every real bound along the specified direction. Saying \(P(x)\to-\infty\) means that \(P(x)\) eventually falls below every real bound.
The Degree-and-Sign Rule
For \(x\to+\infty\), \(x^n\) is positive for every positive integer \(n\). Thus the leading term’s sign at the positive end is the sign of \(a_n\). At the negative end, the parity of \(n\) also matters: an even power of a negative number is positive, while an odd power is negative.
| Degree | Leading coefficient | As \(x\to+\infty\) | As \(x\to-\infty\) |
|---|---|---|---|
| Even, positive | \(a_n>0\) | \(P(x)\to+\infty\) | \(P(x)\to+\infty\) |
| Even, negative | \(a_n<0\) | \(P(x)\to-\infty\) | \(P(x)\to-\infty\) |
| Odd, positive | \(a_n>0\) | \(P(x)\to+\infty\) | \(P(x)\to-\infty\) |
| Odd, negative | \(a_n<0\) | \(P(x)\to-\infty\) | \(P(x)\to+\infty\) |
This table applies to nonzero polynomials of positive degree. A nonzero constant polynomial has the same finite value at both ends, and the zero polynomial is identically zero.
Why the Rule Works
The earlier material on leading-term dominance establishes the key idea: as \(|x|\) grows, the leading term determines the polynomial’s end behavior. The following factorization shows why this leads to the degree-and-sign rule.
Each term inside the parentheses other than \(1\) tends to zero as \(x\to+\infty\) or \(x\to-\infty\). There are only finitely many such terms, so the entire parenthetical factor tends to \(1\). In particular, it is positive for all sufficiently large \(|x|\). Therefore, far enough toward either end, the sign of \(P(x)\) is the sign of \(a_nx^n\).
Also, the absolute value of the parenthetical factor is eventually greater than \(1/2\). Consequently, for sufficiently large \(|x|\),
Because \(n\) is positive, the right-hand side grows without bound as \(|x|\to\infty\). Thus the magnitude of \(P(x)\) becomes unbounded, and its eventual sign determines whether its limit is \(+\infty\) or \(-\infty\). Finally, \(x^n\) is positive at both ends when \(n\) is even, while for odd \(n\) it is positive at \(+\infty\) and negative at \(-\infty\). This proves the rule.
Worked Examples
Worked Example: An Even Degree with a Positive Leading Coefficient
Find the limits of \(P(x)=4x^4-6x^2+5\) as \(x\to+\infty\) and \(x\to-\infty\).
The highest power with a nonzero coefficient is \(x^4\), so the degree is \(4\), which is even. The leading coefficient is \(4>0\). Factoring out the leading term gives
As \(x\) tends to either infinity, the terms \(-3/(2x^2)\) and \(5/(4x^4)\) tend to zero. The parenthetical factor tends to \(1\) and is eventually positive. The factor \(4x^4\) is positive at both ends and grows without bound. Therefore,
Check: Dividing by the leading term gives \(P(x)/(4x^4)=1-3/(2x^2)+5/(4x^4)\to1\). Thus the polynomial eventually has the same positive sign and unbounded magnitude as \(4x^4\), confirming both limits.
Worked Example: An Odd Degree with a Negative Leading Coefficient
Find both end limits of \(Q(x)=-3x^5+2x^3-4x+6\).
The degree is \(5\), which is odd, and the leading coefficient is \(-3<0\). Factor out the leading term:
The parenthetical factor tends to \(1\), so it is eventually positive. When \(x\to+\infty\), \(-3x^5\to-\infty\). When \(x\to-\infty\), \(x^5\to-\infty\), so \(-3x^5\to+\infty\). Hence
Check: The quotient \(Q(x)/(-3x^5)=1-2/(3x^2)+4/(3x^4)-2/x^5\) tends to \(1\). The polynomial therefore has the leading term’s eventual sign and unbounded magnitude at both ends, as the stated limits require.
Worked Example: A Zero Coefficient at the Stated Highest Power
Determine the degree and end limits of \(R(x)=0x^6+5x^2-7x+1\).
The coefficient of \(x^6\) is zero, so the \(x^6\) term contributes nothing. The highest power with a nonzero coefficient is \(x^2\); therefore the degree is \(2\), and the leading coefficient is \(5\). Factor the actual leading term:
The parenthetical factor tends to \(1\) at either end and is eventually positive. Since \(5x^2\) is positive and grows without bound in both directions,
Check: \(R(x)/(5x^2)=1-7/(5x)+1/(5x^2)\to1\). The quotient confirms that the nonzero quadratic term—not the displayed zero \(x^6\) term—controls the end behavior.
Worked Example: A Constant Polynomial
Find the limits of \(S(x)=-8\) at both infinities.
This is a nonzero constant polynomial, so its value does not change as \(x\) changes. In particular, for every real \(x\), \(S(x)=-8\). Therefore,
Check: Substituting any input, for example \(x=10\) or \(x=-10\), gives \(S(x)=-8\). The constant value is the limit at both ends. The zero polynomial similarly has limit \(0\) at both ends.
A Common Sign Mistake
A frequent error is to look only at the leading coefficient when \(x\to-\infty\). For example, someone might reason that the leading coefficient of \(-4x^3+2x\) is negative, and incorrectly conclude that the polynomial tends to \(-\infty\) at both ends.
The missing factor is \(x^3\). At the negative end, \(x^3\) is negative, so the product \(-4x^3\) is positive. More explicitly, set \(x=-t\), where \(t\to+\infty\):
At the positive end, the leading term \(-4x^3\) is negative and unbounded, so the polynomial tends to \(-\infty\). Thus the correct pair is
The correction is to determine the sign of \(x^n\) at the end under consideration before combining it with the leading coefficient.
Why End Behavior Matters
End behavior gives a compact description of what a polynomial does far from the origin. It can reveal whether the two ends point in the same direction or opposite directions, and whether the polynomial grows upward or downward at each end. The degree identifies the parity pattern; the leading coefficient determines the direction at the positive end and, with parity, at the negative end.
This classification concerns sufficiently large positive or negative inputs. It does not say that every value of the polynomial has the same sign. For instance, lower-degree terms can cause the polynomial to cross the horizontal axis or change direction at finite inputs. They simply cannot overturn the leading term’s effect arbitrarily far out.
Check Your Understanding
For each question, use the degree and leading coefficient, and remember to account for the sign of an odd power at negative infinity.
- What are the limits of \(2x^6-3x+1\) as \(x\to+\infty\) and \(x\to-\infty\)?
- What are the limits of \(-5x^3+x^2-4\) at both infinities?
- What is the degree of \(0x^4-7x^2+2x\), and what are its two end limits?
- Why does factoring out the leading term help determine a polynomial’s end behavior?
- What are the limits of the constant polynomial \(9\) at positive and negative infinity?
