Unit 1: Limits and Continuity
Work through these 16 topics in order. Clear the gatekeeper and the rest of the course opens up.
1.1 Change at an instant
Average speed over a stretch of time is easy — distance over time.
1.2 Defining limits and notation
A limit answers “where is the function headed as the input closes in on a point?” — not necessarily where it lands at the point itself.
1.3 Estimating limits from graphs
Put your finger on the curve and slide toward x=c from the left, then from the right.
1.4 Estimating limits from tables
Feed the function inputs that close in on c from both sides and watch the outputs settle toward a number.
1.5 Limit laws (algebraic properties)
When each piece of an expression has a limit, the limit passes through sums, products, and quotients — so you can often just substitute.
1.6 Limits by algebraic manipulation
When substitution gives 0/0, the expression is hiding a removable factor.
1.7 Selecting a procedure for a limit
Try the cheapest tool first (substitution).
1.8 The Squeeze (Sandwich) Theorem
If a wiggly function is trapped between two functions that meet at the same height, it is forced to that height too.
1.9 Connecting representations of a limit
A real limit is the same whether you read it from a graph, a table, or algebra — the representations must agree.
1.10 Types of discontinuities
A break in a graph comes in three flavors: a hole, a jump, or a blow-up.
1.11 Continuity at a point
A function is continuous at a point if you can draw through it without lifting your pen — no hole, jump, or blow-up there.
1.12 Continuity over an interval
Continuous on an interval means unbroken across the whole stretch — with endpoints only needing the one-sided approach that stays inside.
1.13 Removing removable discontinuities
If a graph has a single hole, you can plug it — define the missing point to be exactly the limit.
1.14 Infinite limits and vertical asymptotes
Near certain inputs a function shoots up or down without bound — the graph races along a vertical line.
1.15 Limits at infinity and horizontal asymptotes
Zoom out: what height does the function level off to as x runs far right or far left?
1.16 The Intermediate Value Theorem
An unbroken curve that starts below a height and ends above it must cross that height somewhere in between.