1Unit 1: Limits and Continuity

Average speed over a stretch of time is easy — distance over time.

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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.

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Put your finger on the curve and slide toward $x=c$ from the left, then from the right.

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Feed the function inputs that close in on $c$ from both sides and watch the outputs settle toward a number.

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When each piece of an expression has a limit, the limit passes through sums, products, and quotients — so you can often just substitute.

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When substitution gives $0/0$, the expression is hiding a removable factor.

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Try the cheapest tool first (substitution).

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If a wiggly function is trapped between two functions that meet at the same height, it is forced to that height too.

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A real limit is the same whether you read it from a graph, a table, or algebra — the representations must agree.

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A break in a graph comes in three flavors: a hole, a jump, or a blow-up.

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A function is continuous at a point if you can draw through it without lifting your pen — no hole, jump, or blow-up there.

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Continuous on an interval means unbroken across the whole stretch — with endpoints only needing the one-sided approach that stays inside.

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If a graph has a single hole, you can plug it — define the missing point to be exactly the limit.

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Near certain inputs a function shoots up or down without bound — the graph races along a vertical line.

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Zoom out: what height does the function level off to as $x$ runs far right or far left?

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An unbroken curve that starts below a height and ends above it must cross that height somewhere in between.

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