Limits
Five units on the idea that holds calculus together. You start with the two questions that led to calculus, the slope of a curve and the area under it, then estimate limits from tables and graphs, meet one-sided and infinite limits, use the limit laws to compute exact values, and finish with continuity. Each unit has a calculus lab where you move x toward a point or shrink a secant line and watch the numbers settle.
Before you start
Limits are computed on functions such as (x² − 4)/(x − 2), so you need function notation from Functions and the factoring practised in Quadratics. Square roots and slope of a line also appear early. The calculus lab handles the long decimal tables, so you can focus on what the numbers approach.
When you finish
You will be able to explain how secant lines on y = x² close in on a tangent, estimate a limit from a table where 3.999 and 4.001 approach 4, and tell one-sided limits apart. You can find vertical asymptotes, turn 0/0 into an exact 1/6 with the limit laws, apply the squeeze theorem, and show that x³ − x − 1 crosses zero between 1 and 2.
- 1The two questions behind calculusOn the curve y = x², the line through (1, 1) and (2, 4) has slope 3. Move the second point to (1.5, 2.25) and the slope drops to 2.5.
- 2The limit of a functionThe function (x² − 4)/(x − 2) has no value at x = 2, yet at x = 1.999 it gives 3.999 and at 2.001 it gives 4.001.
- 3One-sided and infinite limitsJust left of 2, |x − 2|/(x − 2) is always −1; just right of 2 it is always 1. The two sides never agree.
- 4The limit lawsPut x = 9 into (√x − 3)/(x − 9) and you get 0/0. Multiply top and bottom by √x + 3 and the limit turns out to be exactly 1/6.
- 5Continuityf(x) = x³ − x − 1 is −1 at x = 1 and 5 at x = 2. Because it has no breaks, it must cross zero somewhere in between.
Sources for this course (5)
- OpenStax, Calculus Volume 1 , chapter 2 Limits: 2.1 A Preview of Calculus (tangent problem and area problem; secant slopes through (1, 1) on f(x) = x² with (2, 4) giving 3 and (3/2, 9/4) giving 2.5; rock s(t) = −16t² + 64 with average velocities −15.84 and −16.16 near t = 0.5; three rectangles under x² + 1 on [0, 3] giving 8: openstax.org/books/calculus-volume-1/pages/2-1-a-preview-of-calculus)
- 2.2 The Limit of a Function ((x² − 4)/(x − 2) → 4 at 2, (sin x)/x → 1 at 0 from a table, sin(1/x) has no limit at 0, one-sided limits of |x − 2|/(x − 2), infinite limits and vertical asymptotes, the formal definition appearing only in the late 19th century: .../pages/2-2-the-limit-of-a-function)
- 2.3 The Limit Laws (sum, difference, constant multiple, product, quotient, power and root laws; direct substitution; factoring and rationalizing 0/0 forms; squeeze theorem: .../pages/2-3-the-limit-laws)
- 2.4 Continuity (three conditions; removable, jump and infinite discontinuities; Intermediate Value Theorem: .../pages/2-4-continuity)
- 2.5 The Precise Definition of a Limit (.../pages/2-5-the-precise-definition-of-a-limit).
Checked: 2026-10-17