L’Hôpital’s Rule
Resolve 0/0 or ∞/∞ with derivatives.
Key idea
Resolve 0/0 or ∞/∞ with derivatives.
In depth
For 0/0 or ∞/∞ limits, lim f/g = lim f′/g′ if the latter exists. It compares how fast numerator and denominator are changing. Other indeterminate forms must first be rewritten.
Worked example
lim (sin x)/x as x → 0 = lim cos x/1 = 1.
Formula
Real-world uses
- Evaluating limits in physics and chemistry where direct substitution gives 0/0
Common mistake
Using it when the form is not indeterminate, or differentiating the quotient as a whole.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is L’Hôpital’s Rule?
Resolve 0/0 or ∞/∞ with derivatives. It is a topic at the expert level in Calculus.
What is the formula for L’Hôpital’s Rule?
The key formula is lim f/g = lim f′/g′.
Where is L’Hôpital’s Rule used in real life?
Real-world examples: Evaluating limits in physics and chemistry where direct substitution gives 0/0.
What should I learn before L’Hôpital’s Rule?
Make sure you are comfortable with Rolle’s Theorem, the previous topic in Calculus.
What is a common mistake with L’Hôpital’s Rule?
Using it when the form is not indeterminate, or differentiating the quotient as a whole.
How can I practise L’Hôpital’s Rule?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after L’Hôpital’s Rule?
Move on to Curve Sketching next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.