Calculus Applications of differentiation

L'Hôpital's rule

Q: What is L'Hopital's rule?

For a limit of the form 0 over 0 or infinity over infinity, the limit of f over g equals the limit of f prime over g prime: replace top and bottom by their derivatives.

Q: When can you use L'Hopital's rule?

Only when direct substitution gives an indeterminate form 0 over 0 or infinity over infinity. Check the form before applying it.

Q: Can you apply L'Hopital's rule more than once?

Yes. If the new limit is still 0 over 0 or infinity over infinity, apply the rule again until the form is determinate.

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L'Hopital's Rule

California Calculus • Standard 8.0 • Applications of Differentiation

L'Hopital's Rule is a topic in Applications of Differentiation in the California Calculus Standards. It is aligned to Standard 8.0, which requires students to know and apply L'Hopital's rule, alongside Rolle's Theorem and the Mean Value Theorem.

L'Hopital's rule evaluates an indeterminate limit of the form \(\dfrac{0}{0}\) or \(\dfrac{\infty}{\infty}\) by replacing the numerator and denominator with their derivatives and taking the limit again.

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Theory

L'Hopital's rule evaluates a limit that gives an indeterminate form \(\dfrac{0}{0}\) or \(\dfrac{\infty}{\infty}\): replace the top and bottom by their derivatives and take the limit again. Repeat if the form persists.

Some limits give an indeterminate form when you substitute — \(\dfrac{0}{0}\) or \(\dfrac{\infty}{\infty}\) — where the answer is not determined by the form alone.

L'Hopital's rule says that for such a form,

\[\lim_{x\to a}\dfrac{f(x)}{g(x)}=\lim_{x\to a}\dfrac{f'(x)}{g'(x)},\]

provided the right-hand limit exists. You differentiate the numerator and denominator separately (this is not the quotient rule).

Key idea: the rule applies only to \(\dfrac{0}{0}\) and \(\dfrac{\infty}{\infty}\). Always check the form first; if it is already a number, do not use the rule.

Both \(f\) and \(g\) vanish at \(a\): a \(\dfrac{0}{0}\) form.

After the rule, the ratio approaches a finite limit.

The rule (for an indeterminate form):

\[\lim_{x\to a}\dfrac{f(x)}{g(x)}=\lim_{x\to a}\dfrac{f'(x)}{g'(x)}\]

limit of f over g equals limit of f prime over g prime

Differentiate top and bottom separately. This is not the quotient rule — there is no \(g^2\) in the denominator.

How to apply L'Hopital's rule

Check the form: substitute to confirm \(\dfrac{0}{0}\) or \(\dfrac{\infty}{\infty}\).
Differentiate the numerator and denominator separately.
Take the limit again; if it is still indeterminate, repeat the rule.

Example 1 — A 0/0 form

Evaluate \(\displaystyle\lim_{x\to 0}\dfrac{\sin x}{x}\).

Solution

Substituting gives \(\dfrac{0}{0}\); differentiate top and bottom.

\(\lim_{x\to 0}\dfrac{\sin x}{x}\)	\(=\)	\(\lim_{x\to 0}\dfrac{\cos x}{1}\)
	\(=\)	\(1\)

limit equals 1 by L'Hopital's rule

Example 2 — Another 0/0

Evaluate \(\displaystyle\lim_{x\to 0}\dfrac{e^{x}-1}{x}\).

Solution

This is \(\dfrac{0}{0}\); apply the rule.

\(\lim_{x\to 0}\dfrac{e^{x}-1}{x}\)	\(=\)	\(\lim_{x\to 0}\dfrac{e^{x}}{1}\)
	\(=\)	\(1\)

limit equals 1

Example 3 — An \(\infty/\infty\) form

Evaluate \(\displaystyle\lim_{x\to\infty}\dfrac{x}{e^{x}}\).

Solution

This is \(\dfrac{\infty}{\infty}\); differentiate top and bottom.

\(\lim_{x\to\infty}\dfrac{x}{e^{x}}\)	\(=\)	\(\lim_{x\to\infty}\dfrac{1}{e^{x}}\)
	\(=\)	\(0\)

limit equals 0

Example 4 — Apply it twice

Evaluate \(\displaystyle\lim_{x\to 0}\dfrac{1-\cos x}{x^2}\).

Solution

\(\dfrac{0}{0}\) once gives \(\dfrac{\sin x}{2x}\), still \(\dfrac{0}{0}\) — apply the rule again.

\(\lim_{x\to 0}\dfrac{1-\cos x}{x^2}\)	\(=\)	\(\lim_{x\to 0}\dfrac{\sin x}{2x}\)
	\(=\)	\(\lim_{x\to 0}\dfrac{\cos x}{2}=\dfrac{1}{2}\)

limit equals one half after applying twice

Common pitfalls

Only for indeterminate forms. If substitution already gives a number (or \(\dfrac{k}{0}\)), L'Hopital does not apply.

It is not the quotient rule. Differentiate top and bottom separately; there is no denominator-squared.

Recheck the form each time. After one application the limit may still be \(\dfrac{0}{0}\) — apply the rule again, or stop once it is determinate.

Frequently asked questions

What is L'Hopital's rule?

For a limit of the form \(\dfrac{0}{0}\) or \(\dfrac{\infty}{\infty}\), \(\lim\dfrac{f}{g}=\lim\dfrac{f'}{g'}\): replace top and bottom by their derivatives.

When can you use L'Hopital's rule?

Only when direct substitution gives an indeterminate form \(\dfrac{0}{0}\) or \(\dfrac{\infty}{\infty}\). Check the form before applying it.

Is L'Hopital's rule the same as the quotient rule?

No. You differentiate the numerator and denominator separately; you do not use the quotient-rule formula.

Can you apply L'Hopital's rule more than once?

Yes. If the new limit is still \(\dfrac{0}{0}\) or \(\dfrac{\infty}{\infty}\), apply the rule again until the form is determinate.

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