AcademyApplications Of Derivatives
Academy
L'Hopital's Rule
Level 1 - Calculus topic page in Applications Of Derivatives.
Principle
L'Hopital's rule evaluates certain indeterminate limits by comparing derivatives of the numerator and denominator. It applies only after the limit has been shown to have an allowed indeterminate form.
Notation
\(0/0\)
zero over zero indeterminate form
\(\infty/\infty\)
infinite over infinite indeterminate form
\(f'(x),g'(x)\)
derivatives of numerator and denominator
The Core Method
If \(f(x)\) and \(g(x)\) both approach \(0\), or both grow without bound, and the differentiability conditions hold, then
L'Hopital's rule
\[\lim_{x\to a}\frac{f(x)}{g(x)}=\lim_{x\to a}\frac{f'(x)}{g'(x)}\]
provided the derivative limit exists. Do not use the rule for ordinary finite non-zero quotients.
Worked Cases
Question
Evaluate \(\lim_{x\to0}\frac{e^x-1}{x}\).
Answer
Substitution gives \((1-1)/0=0/0\), so the form is allowed. Differentiate numerator and denominator: derivative of \(e^x-1\) is \(e^x\), and derivative of \(x\) is \(1\). The limit becomes \(\lim_{x\to0}e^x=1\).
Examples
Question
Evaluate \(\lim_{x\to\infty}\frac{x}{e^x}\).
Answer
Both numerator and denominator grow without bound. Apply L'Hopital's rule: \(\lim_{x\to\infty}\frac{x}{e^x}=\lim_{x\to\infty}\frac{1}{e^x}=0\).