Unit 8: Limits and the definition of the derivative
About this unit
A limit is the value a function approaches near a point, whatever happens at the point itself, and the derivative is the limit of the difference quotient (f(a + h) − f(a))/h as h approaches 0.
By the end, you can: Read a limit as the value a function approaches, even across a hole, then define the derivative as the limit of the difference quotient and work out x² from it.
Modules
5 modulesLearn
- A value you approachArticle · lim f(x) as x → a is the value f(x) gets close to as x gets close to a from both sides; f(a) itself doesn't matter, as the hole in (x² − 1)/(x − 1) at 1 shows.Not started
- The derivative is a limitArticle · f′(a) = lim (f(a + h) − f(a))/h as h → 0, the slope secants approach; for x² the quotient is 2x + h for h ≠ 0, so f′(x) = 2x.Not started
- Close in on the tangentExperiment · Slide h toward 0 on the secant of x² at x = 3 until the secant's slope is within 0.05 of 6, from either side.Not started
- Limits and derivativesReview · A limit is the value approached; the derivative is the limit of the difference quotient; cancel h because h ≠ 0.Not started