Unit 1

Limits & Continuity

10-12% AB & BCCED Unit 1

Foundations

Reference angles, limit definitions, continuity, discontinuities, and limit laws.

The Rule of Four

AP® Mathematical Practice 2 (Connecting Representations) says every problem can be approached through four lenses. We will work limits in all four:

Analytical: algebra and symbols.

Graphical: graphs and geometry.

Numerical: tables and computed values.

Verbal: words and written explanation.

→ Tip

Watch for a small tag near each section showing which representation is in play. The strongest students can translate fluently between all four.

Reference Angles

Numerical ·Analytical

You don't need the full unit circle. Memorize first quadrant values only:

For any angle : The denominator tells how many slices in the upper half-circle; the numerator tells how many slices to pass through. Find the reference angle, evaluate, then apply the quadrant sign (ASTC: All Students Take Calculus).

Limit Definition

Verbal ·Analytical

If the limit of

approaches

equals

, then

gets arbitrarily close to

from either side as

approaches

. The function

does not need to be defined at

; even if it is,

does not need to equal

◉ Observe

"From either side" is the whole point. A two-sided limit exists only when the left-hand and right-hand limits both exist and agree.

One-Sided Limits

Analytical ·Graphical

Left:

, from values less than

Right:

, from values greater than

→ Tip

A two-sided limit exists if and only if both one-sided limits exist and are equal.

Continuity at a Point

Analytical ·Verbal

is continuous at if all three conditions hold:

is defined.

exists.

Theorem: Shortcut for Continuity at a Point

is continuous at

if and only if

One equation, three conditions. Writing

on the right requires

to be defined; the limit existing requires both one-sided limits to exist and agree; setting them equal handles the third condition.

Types of Discontinuities

Graphical ·Verbal

Type	What Happens	Graph
Removable	Limit exists but , or undefined	Hole
Jump	One-sided limits exist but aren't equal	Step / break
Infinite	Limit is	Vertical asymptote

Properties of Limits

Analytical

Let

and

be real numbers,

a positive integer, and

functions with

and

Property	Statement
Scalar Multiple
Sum / Difference
Product
Quotient	, provided
Power
Composite (limit form)

Theorem: Composite Function Theorem

Analytical ·Verbal

and

is continuous at

, then

→ Tip

The property says you can move the limit inside

when it works. The theorem tells you when it works:

has to be continuous at

Special Limits

Analytical

→ Tip

These can be memorized, but each one is a

indeterminate form, so they can also be evaluated with L'Hôpital's Rule (Unit 4).

AP® Calculus AB & BC · Unit 1 Overview · Mr. Brantley