IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus
L’Hôpital’s Rule
Recognise indeterminate 0/0 and ∞/∞ forms and evaluate limits with L’Hôpital’s rule when appropriate.
Learning goal
Evaluate indeterminate-form limits, including repeated use where necessary.
Syllabus link
AA HL: AHL 5.13 · Topic 5 Calculus.
Big idea
L’Hôpital’s rule applies to a quotient limit only after the expression is confirmed to have an appropriate indeterminate form.
Key relationship
\(\lim\frac{f(x)}{g(x)}=\lim\frac{f\'(x)}{g\'(x)}\quad\text{for suitable }0/0\text{ or }\infty/\infty\text{ forms}\)
Core idea
L’Hôpital’s rule applies to a quotient limit only after the expression is confirmed to have an appropriate indeterminate form.
Evaluate indeterminate-form limits, including repeated use where necessary.
Key results and method
- Substitute/inspect the original limit first.
- Confirm \(0/0\) or \(\infty/\infty\).
- Differentiate numerator and denominator separately.
- Repeat only if the new quotient remains indeterminate.
Worked example
Evaluate \(\lim_{x\to0}\frac{e^x-1}{x}\).
- Direct substitution gives \(0/0\).
- Apply L’Hôpital: numerator derivative \(e^x\), denominator derivative \(1\).
- Now substitute \(x=0\).
Practice
Try these questions before moving on.
- Evaluate a trigonometric 0/0 limit.
- Evaluate an ∞/∞ rational-exponential limit.
- Find a limit requiring two applications of the rule.
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