Introduction
As \(x\) becomes large, different familiar functions grow at very different rates. A logarithm increases without bound, but very slowly. A positive power of \(x\) grows faster than any fixed power of \(\ln x\). An exponential with positive growth rate eventually grows faster than any fixed power of \(x\).
These comparisons are useful because they let us evaluate limits without estimating large numerical values. L’Hôpital’s Rule turns certain ratios into ratios of derivatives. Applied carefully, it makes the growth comparison precise.
Recognizing the limits we want to compare
The main expressions in this topic have the form \((\ln x)^m/x^p\) or \(x^m/e^{cx}\), where \(m\) is a positive integer and \(p,c>0\). For sufficiently large \(x\), these ratios have the indeterminate form \(\infty/\infty\): both numerator and denominator tend to infinity. That form alone does not determine the limit.
The infinity-over-infinity form of L’Hôpital’s Rule applies when the numerator and denominator are differentiable for all sufficiently large \(x\), the denominator’s derivative is nonzero there, both original functions tend to infinity, and the ratio of their derivatives has a limit. Under those conditions, the original ratio has the same limit as the ratio of derivatives. When applying the rule repeatedly, check that the new ratio still meets the required conditions before using it again.
For logarithmic powers, the substitution \(t=\ln x\) is particularly helpful. As \(x\to\infty\), \(t\to\infty\), and \(x^p=e^{pt}\). Thus a logarithm-versus-power comparison becomes a power-versus-exponential comparison in \(t\).
Worked Example: A fourth power of a logarithm over a cubic
Worked Example: A fourth power of a logarithm over a cubic
Evaluate \(\displaystyle\lim_{x\to\infty}\frac{(\ln x)^4}{x^3}\).
Set \(t=\ln x\). Then \(t\to\infty\), \((\ln x)^4=t^4\), and \(x^3=e^{3t}\). The limit becomes
Initially, numerator and denominator both tend to infinity, and the denominator’s derivative is nonzero. Apply L’Hôpital’s Rule:
At each of the first three applications, the numerator and denominator still tend to infinity, and the denominator’s derivative is nonzero. After the fourth differentiation, the numerator is constant while \(e^{3t}\to\infty\), so the final limit is zero. Substituting back, \(\displaystyle\lim_{x\to\infty}\frac{(\ln x)^4}{x^3}=0\). This confirms that the cubic eventually outgrows the fourth power of the logarithm.
Why logarithmic powers are dominated by powers of \(x\)
The example illustrates a general result. The proof uses the substitution \(t=\ln x\) and applies L’Hôpital’s Rule to a power of \(t\) divided by an exponential. In the intermediate steps, both parts of the ratio tend to infinity; after the last differentiation, the numerator is constant and the denominator still tends to infinity.
To prove the result, set \(t=\ln x\). The original ratio becomes \(t^m/e^{pt}\). After \(k\) differentiations of numerator and denominator, where \(0\leq k<m\), the ratio is
For each \(k<m\), the power \(t^{m-k}\) and the exponential \(e^{pt}\) both tend to infinity. The denominator derivative is nonzero because \(p>0\). After \(m\) applications, the ratio of derivatives is \(m!/(p^m e^{pt})\), which tends to zero since \(e^{pt}\to\infty\). L’Hôpital’s Rule therefore gives the same limit for \(t^m/e^{pt}\). Since \(t=\ln x\), this proves the stated limit.
This theorem shows that no fixed positive power of \(\ln x\) can keep pace with any fixed positive power of \(x\). The values of \(m\) and \(p\) can differ: the logarithm may be raised to a larger fixed integer power than the power of \(x\) in the denominator.
Worked Example: A polynomial over an exponential
Worked Example: A polynomial over an exponential
Evaluate \(\displaystyle\lim_{x\to\infty}\frac{x^4}{e^{2x}}\).
Both numerator and denominator tend to infinity. Since the derivative of the denominator, and of each later exponential denominator, is nonzero, L’Hôpital’s Rule can be applied while the numerator also tends to infinity:
Each line differentiates the numerator and denominator of the preceding ratio; the displayed coefficients are the resulting derivative values. In the last ratio, the numerator is constant and the denominator tends to infinity, so its limit is zero. Thus \(\displaystyle\lim_{x\to\infty}x^4/e^{2x}=0\).
Why exponentials dominate polynomials
The same reasoning proves the comparison for any nonnegative integer power and any positive exponential growth rate.
For \(0\leq k<m\), after \(k\) differentiations of numerator and denominator the ratio is
At each of those stages, the numerator and denominator tend to infinity, and the denominator’s derivative is nonzero. After \(m\) applications, the ratio is \(m!/(c^m e^{cx})\), which tends to zero. L’Hôpital’s Rule therefore proves the result. If the polynomial has degree zero, the ratio is a constant divided by \(e^{cx}\), which also tends to zero directly.
The exponential need not be written with base \(e\). If \(a>1\), then \(a^x=e^{x\ln a}\), and \(c=\ln a>0\). The theorem consequently gives \(\lim_{x\to\infty}x^m/a^x=0\).
Worked Example: A polynomial over an exponential with another base
Worked Example: A polynomial over an exponential with another base
Evaluate \(\displaystyle\lim_{x\to\infty}\frac{x^3}{7^x}\).
Rewrite the denominator using \(7^x=e^{x\ln 7}\). Since \(7>1\), \(\ln 7>0\), so both \(x^3\) and \(e^{x\ln 7}\) tend to infinity. L’Hôpital’s Rule applies:
The first two derivative ratios still have numerator and denominator tending to infinity, and each denominator derivative is nonzero. In the last ratio, the numerator is constant and the denominator tends to infinity because \(\ln 7>0\). Hence the limit is zero, as required.
The growth hierarchy
The two results combine to give the hierarchy, for fixed positive integers \(m,n\) and positive \(p,c\):
These limits express a comparison of growth rates, not a claim that logarithms or polynomials remain bounded. Both can grow without bound. The point is that their ratios to the faster-growing functions approach zero. For example, a logarithmic power is eventually small relative to a positive power of \(x\), while a polynomial is eventually small relative to a positive-rate exponential.
The results also help evaluate more complicated ratios. If a numerator contains a fixed power of \(\ln x\) and the denominator contains a positive power of \(x\), the first theorem may settle the limit. If a polynomial is divided by an exponential with base greater than one, rewrite that exponential as \(e^{cx}\) and use the second theorem.
A common mistake: differentiating the quotient itself
L’Hôpital’s Rule does not say to differentiate the quotient as a single function. It says to compare the derivatives of the numerator and denominator, when its conditions are satisfied. For instance, consider \((\ln x)^2/x\). A mistaken calculation might write
This is not the derivative of the quotient. The quotient rule gives
To use L’Hôpital’s Rule for the limit of the original ratio, differentiate its numerator and denominator separately instead:
The last limit is zero because \(\ln x/x\to 0\), as established by the logarithm-versus-power comparison. The distinction matters: the quotient rule finds the derivative of a quotient, whereas L’Hôpital’s Rule compares the derivatives of its two parts.
Summary
For limits comparing the standard growth rates at infinity, L’Hôpital’s Rule gives a systematic method. A substitution \(t=\ln x\) converts logarithmic powers over powers of \(x\) into powers of \(t\) over exponentials. Repeated differentiation then shows that the ratio tends to zero. For polynomial powers over exponentials, repeated differentiation reduces the polynomial to a constant while the exponential continues to grow.
Before applying the rule, verify the indeterminate form, differentiability, and nonzero denominator derivative. After every application, verify that another application is justified rather than assuming the original conditions automatically continue to hold.
Check Your Understanding
For each problem, identify the growth comparison and give a reasoned evaluation of the limit.
- Evaluate \(\displaystyle\lim_{x\to\infty}\frac{(\ln x)^3}{x^2}\). Which substitution turns it into a power-over-exponential limit?
- Evaluate \(\displaystyle\lim_{x\to\infty}\frac{x^2}{e^{4x}}\). How many applications of L’Hôpital’s Rule are needed to reach a constant numerator?
- Explain why \(\displaystyle\lim_{x\to\infty}x^5/3^x=0\), and state why \(\ln 3\) has the sign needed in the argument.
- State the conditions that must be checked before applying L’Hôpital’s Rule to an infinity-over-infinity limit.
- What is the difference between differentiating a quotient and applying L’Hôpital’s Rule to a quotient’s limit?
