Rational Functions at Infinity
A rational function is a quotient of polynomials. To find its behavior as the input grows without bound, the degrees of the numerator and denominator provide a useful first guide. When both polynomials grow without bound in magnitude, the quotient has the indeterminate form infinity over infinity, so L'Hôpital's Rule can also be useful.
For these limits, the two methods reinforce one another. L'Hôpital's Rule differentiates the polynomials, while degree comparison predicts the result and provides a check. At negative infinity, the leading terms also determine the sign: an odd power of a negative input is negative, whereas an even power is positive.
When the Infinity-Over-Infinity Form Applies
Let \(P\) and \(Q\) be polynomials, and consider \(P(x)/Q(x)\) as \(x\) tends to \(+\infty\) or \(-\infty\). If both polynomials have positive degree, then their absolute values tend to infinity at either endpoint. The quotient therefore has the infinity-over-infinity form, regardless of whether either polynomial tends to positive or negative infinity.
The version of L'Hôpital's Rule used here applies on a ray extending toward the endpoint. If \(P\) and \(Q\) are differentiable there, \(|P(x)|\) and \(|Q(x)|\) tend to infinity, \(Q'(x)\ne 0\) sufficiently far along that ray, and \(P'(x)/Q'(x)\) has a limit (finite or infinite), then \(P(x)/Q(x)\) has the same limit. For a polynomial quotient, each application differentiates the numerator and denominator separately.
When applying the rule more than once, the quotient of derivatives must still be in a form to which the rule applies. For polynomials, this can be checked using their degrees: a derivative of a polynomial of degree \(d\ge 1\) has degree \(d-1\), and its absolute value tends to infinity if that degree is positive. The derivatives of a nonconstant polynomial are nonzero sufficiently far along either ray, since their leading terms dominate there.
The Degree Comparison
Write the numerator and denominator in terms of their degrees and leading coefficients:
Here \(a_m\ne 0\), \(b_n\ne 0\), and \(m,n\) are nonnegative integers. If \(m,n\ge 1\), the quotient starts in the infinity-over-infinity form. The degree comparison gives its limit directly.
- If \(m<n\), then \(P(x)/Q(x)\to 0\) as \(x\to+\infty\) or \(x\to-\infty\).
- If \(m=n\), then \(P(x)/Q(x)\to a_m/b_n\) at either endpoint.
- If \(m>n\), the magnitude of \(P(x)/Q(x)\) tends to infinity. At \(+\infty\), its sign is the sign of \(a_m/b_n\). At \(-\infty\), its sign is the sign of \((a_m/b_n)(-1)^{m-n}\).
To justify the result, divide the numerator and denominator by \(x^n\). For \(x\ne 0\), this gives
Each negative power tends to zero as \(x\) tends to either infinity. The fraction on the right therefore tends to \(a_m/b_n\), which is nonzero. If \(m<n\), the factor \(x^{m-n}\) tends to zero in magnitude, so the whole product tends to zero. If \(m=n\), that factor is \(1\), leaving the limit \(a_m/b_n\). If \(m>n\), the factor's magnitude tends to infinity. At negative infinity, its sign is determined by whether the integer \(m-n\) is even or odd. This proves the comparison and its sign rule.
Worked Example: Equal Degrees at Positive Infinity
Worked Example: Equal Degrees at Positive Infinity
Evaluate \(\displaystyle \lim_{x\to+\infty}\frac{6x^2-3x+4}{2x^2+7x-1}\) using L'Hôpital's Rule, and check the result by degree comparison.
The numerator and denominator both have degree \(2\), so each tends to \(+\infty\) as \(x\to+\infty\). The quotient is in the infinity-over-infinity form. The denominator's derivative is \(4x+7\), which is nonzero for all sufficiently large \(x\). L'Hôpital's Rule gives
The differentiated numerator and denominator both tend to \(+\infty\), so the derivative quotient is again in the infinity-over-infinity form. Divide its numerator and denominator by \(x\):
The original polynomials have equal degree, and their leading-coefficient ratio is \(6/2=3\). The degree comparison agrees. As a check, the original quotient divided by \(x^2\) in numerator and denominator tends to \(6/2=3\), as required.
Using L'Hôpital's Rule When the Degrees Differ
When the degrees differ, repeated differentiation can expose the same conclusion as the degree comparison. If the numerator has lower degree, differentiating eventually reduces it to a constant while the denominator still has positive degree. If the degrees are equal, differentiating both polynomials the same number of times eventually produces constants. If the numerator has higher degree, it remains a nonconstant polynomial after the denominator has been reduced to a constant.
These observations describe what happens for polynomial quotients; they do not replace the conditions of L'Hôpital's Rule. At each use, the current numerator and denominator must have absolute values tending to infinity, and the current denominator's derivative must be nonzero sufficiently far along the ray. In the examples below, the derivative degrees make these conditions explicit.
Worked Example: Lower-Degree Numerator at Negative Infinity
Evaluate \(\displaystyle \lim_{x\to-\infty}\frac{4x^2-1}{2x^5+3x}\) using L'Hôpital's Rule.
At negative infinity, \(4x^2-1\to+\infty\), while \(2x^5+3x\to-\infty\). Both absolute values tend to infinity, so the quotient is in the infinity-over-infinity form. The denominator's derivative is \(10x^4+3\), which is positive and nonzero. Applying L'Hôpital's Rule once gives
Here \(8x\to-\infty\), and \(10x^4+3\to+\infty\), so the derivative quotient is still in the infinity-over-infinity form. Its denominator's derivative is \(40x^3\), which is nonzero for \(x<0\). Apply the rule again:
For the final limit, \(40x^3\to-\infty\), so the quotient tends to zero. The numerator's degree is \(2\), less than the denominator's degree \(5\); degree comparison also predicts a limit of zero. The sign near negative infinity is negative, but the quotient still tends to zero.
Worked Example: Higher-Degree Numerator at Negative Infinity
Evaluate \(\displaystyle \lim_{x\to-\infty}\frac{3x^5-x^2+1}{2x^2+4}\) using L'Hôpital's Rule and determine the sign of the result.
The numerator tends to \(-\infty\), and the denominator tends to \(+\infty\). Their absolute values both tend to infinity. The derivative of the denominator, \(4x\), is nonzero along the negative ray. One application gives
The numerator \(15x^4-2x\) tends to \(+\infty\), while \(4x\) tends to \(-\infty\), so their absolute values still tend to infinity. The derivative of \(4x\) is \(4\), which is nonzero. Applying the rule a second time gives
The numerator's degree exceeds the denominator's by \(3\). The leading-term comparison is \((3/2)x^3\), which tends to \(-\infty\) at negative infinity because \(x^3\to-\infty\). This confirms both the magnitude and the sign of the result.
A Common Sign Mistake
A common error is to compare degrees and conclude that a higher-degree numerator always gives positive infinity. That conclusion ignores the direction of approach and the parity of the degree difference.
Worked Example: Correcting the Sign at Negative Infinity
Consider \(\displaystyle \lim_{x\to-\infty}\frac{x^3+1}{x^2+2}\). A wrong calculation might say: “The numerator has degree \(3\), the denominator has degree \(2\), so the quotient tends to \(+\infty\).” The degree comparison does imply an unbounded magnitude, but it does not support that sign.
Compare the leading terms by dividing through by \(x^2\):
The fraction multiplying \(x\) tends to \(1\), while \(x\to-\infty\). Therefore the limit is \(-\infty\), not \(+\infty\). Equivalently, the degree difference is \(3-2=1\), which is odd, so its power of \(x\) is negative at negative infinity. This agrees with the rule \((1)(-1)^1=-1\) for the sign.
How to Choose and Check a Method
For a rational-function limit at either infinite endpoint, first identify the degrees and leading coefficients. If both degrees are positive, the quotient has the infinity-over-infinity form and L'Hôpital's Rule is available when its conditions hold. The degree comparison supplies a prediction before any differentiation; after differentiating, use that prediction to check the result.
Record the degrees, leading coefficients, and whether the endpoint is positive or negative infinity.
For L'Hôpital's Rule, verify that the absolute values of both polynomials tend to infinity.
Differentiate numerator and denominator separately, and ensure the conditions remain satisfied before applying the rule again.
Compare degrees for zero, a finite ratio, or unbounded magnitude; at negative infinity, use the parity of the degree difference to check the sign.
Direct degree comparison is often the shorter way to evaluate a polynomial quotient. L'Hôpital's Rule provides another route when the quotient is in the required form. Agreement between the two is a useful check; disagreement usually signals a missed sign, an incorrect derivative, or an invalid use of the rule.
Check Your Understanding
For each question, state the limit and briefly justify it. When asked to use L'Hôpital's Rule, check that its conditions apply.
- Use degree comparison to evaluate \(\displaystyle \lim_{x\to+\infty}\frac{5x^3-2}{10x^3+x}\).
- Evaluate \(\displaystyle \lim_{x\to-\infty}\frac{2x^4+x}{x^6-3}\) and explain why L'Hôpital's Rule can be applied twice.
- Determine whether \(\displaystyle \lim_{x\to-\infty}\frac{-4x^5+1}{2x^2+7}\) is positive or negative infinity. Explain the role of the degree difference.
- For a rational function with numerator degree \(n\) and denominator degree \(n+2\), what does degree comparison predict at either infinite endpoint?
- Why is it not enough to say that a quotient of two polynomials has the infinity-over-infinity form before applying L'Hôpital's Rule?
