Calculus I · Unit 2B · lesson

L'Hopital's Rule for 0/0

Concept

Learning objectives

Apply L'Hopital's Rule to $0/0$ limits and recognize when another method is simpler.

Compare with Algebra and Trigonometric Identities

Explanation

Before the formulas

Indeterminate forms in L'Hopital's Rule for $0/0$ 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

Recheck the form after every transformation

After one application, the new quotient may have an ordinary limit, may still be indeterminate, or may be easier by algebra. Do not differentiate repeatedly by habit. Substitute again and choose the simplest valid next step.

When factoring or a standard trigonometric limit solves the problem more directly, those methods often reveal more structure than L'Hopital's Rule and should remain part of your toolkit.

A $0/0$ limit reflects competing vanishings. L'Hopital's Rule asks which expression vanishes faster by comparing derivatives. In simple algebraic examples, cancellation may reveal the answer more directly; in exponential or trigonometric examples, the derivative comparison can be decisive.

After each application, evaluate the new limit again. The new expression may be ordinary, may remain indeterminate, or may leave the theorem's allowable forms entirely.

Guided walkthrough

A logarithmic limit

Evaluate

\lim_{x\to1}\frac{\ln x}{x-1}.

Answer reveal

Worked solution

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Worked example

Factoring is sometimes faster

For

\lim_{x\to3}\frac{x^2-9}{x-3},

L'Hopital gives $\lim 2x/1=6$ . Factoring gives the same result and reveals the removable hole. Use the method that best exposes the structure.

Worked example

A trig limit requiring two steps of thought

\lim_{x\to0}\frac{1-\cos x}{x^2}

has form $0/0$ . One application gives

\lim_{x\to0}\frac{\sin x}{2x},

still $0/0$ . A second application gives

\lim_{x\to0}\frac{\cos x}{2}=\boxed{\frac12}.

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?

Interactive checklhopital-extra-01

Evaluate $\lim_{x\to0}(1-\cos x)/x^2$ .

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Show hint

Apply L'Hopital twice.

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