L'Hospital's Rule

College/University

Definition

It is also known as L'Hôpital's Rule. Used to evaluate factions' limits to indeterminate expressions of 0/0 and ±∞/±∞. This is carried out through finding limit of the derivatives of the numerator and the denominator. Then we are able to take its limit.

Worked examples

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

The original limit is 0/0, so differentiate top and bottom separately, then evaluate the new limit.

\(\lim_{x \to \infty} \frac{e^x}{x^2} = \lim_{x \to \infty} \frac{e^x}{2x} = \lim_{x \to \infty} \frac{e^x}{2} = \infty\)

An ∞/∞ form; apply L'Hospital's Rule twice until the limit can be evaluated.

\(\lim_{x \to 1} \frac{x^3 - 1}{x - 1} = \lim_{x \to 1} \frac{3x^2}{1} = 3\)

Direct substitution gives 0/0, so take the derivative of numerator and denominator.

Common mistakes

\(\lim_{x \to 0} \frac{\sin x}{x} = \frac{d}{dx}[\sin x] \div \frac{d}{dx}[x]\) → \(\lim_{x \to 0} \frac{(\sin x)'}{(x)'} = \lim_{x \to 0} \frac{\cos x}{1}\) Take derivatives separately, then find the limit of the new fraction — don't divide derivatives before the limit.
\(\lim_{x \to 2} \frac{x^2}{x + 1} = \lim_{x \to 2} \frac{2x}{1}\) → \(\lim_{x \to 2} \frac{x^2}{x + 1} = \frac{4}{3}\) L'Hospital's Rule applies only to 0/0 or ∞/∞; here direct substitution works, so don't differentiate.
\(\lim_{x \to 0} \frac{x^2}{\sin x} = \lim_{x \to 0} \frac{2x}{\cos x} = 0\) → Apply the rule again: \(\lim_{x \to 0} \frac{2}{-\sin x}\) (undefined at 0); limit is 0. If the new limit is still indeterminate, apply L'Hospital's Rule again until it resolves.

Where you'll use it next

You'll use L'Hospital's Rule throughout calculus when evaluating tricky limits in optimization, series convergence tests, improper integrals, and any advanced analysis involving indeterminate forms.

Found in 1 StudyPug lesson

L'Hospital's Rule: Mastering Complex Limits in Calculus

Calculus 1

Unlock the power of L'Hospital's Rule to solve challenging limit problems. Learn how to tackle indeterminate forms, boost your calculus skills, and excel in advanced mathematical analysis.

UniversityUniversity