AP Calculus AB / Contextual Applications of Differentiation
Lesson
Local linear stand-ins and careful 0/0 limits
A tangent line is a simple model. L’Hospital is a tool for stubborn fractions, not a first resort.
Learning goals
- Write a linearization of f at a and use it to estimate f nearby.
- Check an indeterminate form before applying L’Hospital’s rule.
Explanation
Near x = a, f(x) ≈ f(a) + f′(a)(x − a). That line is the linearization. It is excellent close in and worse far away, especially if the graph bends hard. Error talk in this course is usually qualitative unless a prompt asks for more.
If a limit of f/g is 0/0 or ∞/∞, L’Hospital’s rule lets you try f′/g′ instead. It is not a license to differentiate a product in the middle of a sum, and it does not apply to a plain 3/0 blow-up. Algebraic cleanup is still often faster.
Key terms
- Linearization. The tangent-line function used as a local approximation to f.
- Indeterminate form. A limit shape such as 0/0 or ∞/∞ that does not by itself decide the value.
Common mistakes
- Using L’Hospital on a limit that is already a plain number over zero.
- Treating a linearization as exact for any x.
Practice
Original Marlow Works items. Check the answer explanation after you try.
What is lim (x→3) of (x² − 9) / (x − 3)?
Original Marlow Works item — not a College Board question.
Take your time—this is practice, not a test.
Answer explanation
Does L’Hospital apply to lim x→0 (sin x)/x after you already know it is 1, and must you use it?
Answer. The form is 0/0, so the rule could apply, but you are not required to use it if another valid method is available.
The rule is a tool for an indeterminate quotient, not the only legal path.
Related resources
External links with reuse status. Marlow Works is independent and does not copy restricted exam or textbook material.
Official / link only
AP Calculus AB course page
Official eight-unit framework and 2027 exam updates.
College Board · All rights reserved · accessed 2026-10-01
Open sourceReference only — do not copy
OpenStax College Algebra 2e
Algebra reference; current terms include noncommercial and AI-ingestion limits.
OpenStax · Restricted — see publisher page · accessed 2026-10-01
Open source