Calculus I · Unit 2B · lesson

L'Hopital's Rule for infinity/infinity

Concept

Learning objectives

Evaluate $\infty/\infty$ forms and interpret relative growth.

Compare Growth Rates

Explanation

Before the formulas

Indeterminate forms in L'Hopital's Rule for $\infty/\infty$ indicate competition, not ignorance. The expression's pieces may approach values that do not determine the combined limit without more information. Products, differences, and powers must be rewritten before L'Hopital can be considered.

Show the transformation clearly. For a power form, take logarithms, evaluate the logarithmic limit, and then exponentiate. For a difference of large terms, combine or rationalize. The transformation is part of the solution and often reveals a simpler method than L'Hopital.

Explanation

The question is which quantity grows faster

An $\infty/\infty$ form hides a competition between growth rates. Polynomial degree, exponential growth, and logarithmic growth often predict the outcome before calculation. L'Hopital's Rule makes that competition local by comparing derivatives.

Keep signs and one-sided behavior visible. "Infinity" is not a number, and a quotient may approach a finite value, zero, or become unbounded depending on relative growth.

An $\infty/\infty$ form compares competing growth. Differentiation often strips away lower-order behavior until the dominant growth rates become visible. This provides a systematic counterpart to degree comparison and asymptotic reasoning.

The theorem concerns limits, not algebraic equality. Replacing $f/g$ with $f'/g'$ outside a limit is false in general and should feel as suspicious as replacing a journey with its speedometer reading.

Guided walkthrough

Exponential growth beats polynomial growth

Evaluate

\lim_{x\to\infty}\frac{x^3}{e^x}.

Answer reveal

Worked solution

Your setup or answer

Write a real attempt before opening the supplied answer.

Worked example

Logarithm grows more slowly than a power

\lim_{x\to\infty}\frac{\ln x}{x} =\lim_{x\to\infty}\frac{1/x}{1}=0.

Thus $\ln x$ grows more slowly than $x$ .

Method

Growth hierarchy

For positive powers and bases greater than one, the broad hierarchy is

\ln x\ll x^p\ll a^x \qquad(x\to\infty).

Repeated L'Hopital applications make this comparison precise in many quotients.

Application

Comparing algorithmic growth rates

To compare $\ln x$ with $x^{1/2}$ , consider

\lim_{x\to\infty}\frac{\ln x}{\sqrt{x}}.

L'Hopital gives

\lim_{x\to\infty}\frac{1/x}{1/(2\sqrt{x})} =\lim_{x\to\infty}\frac{2}{\sqrt{x}}=0.

Thus logarithmic growth is negligible compared with any positive power in this example. Such comparisons help explain why logarithmic-time algorithms scale so favorably.

After the explanation

Use the section idea

Reading lens

Identify the limiting form before differentiating; transformation and simpler limit laws come before L'Hopital's Rule.

Mental model

The rule compares numerator and denominator growth only for verified zero-over-zero or infinity-over-infinity quotients.

Decision

Evaluate numerator and denominator limits separately, transform nonquotient forms, apply the rule only when justified, then recheck.

Common trap

Using L'Hopital because an expression looks difficult rather than because the required indeterminate quotient has been proved.

Check yourself

Can you name the form at every application and explain why direct substitution or algebra is not already enough?

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