Limits and Continuity

Sixteen topics on what a function is doing as you close in on a point, and what it takes for the graph to be unbroken there. Rates at an instant and the secants that define them, limits read from graphs, tables and algebra, the Squeeze Theorem for functions with no limit of their own, the three kinds of discontinuity and which one can be repaired, continuity at a point and across an interval, vertical and horizontal asymptotes, and the Intermediate Value Theorem.

AB exam 10-15%BC exam 5-10%16 topics

Topics

1.1

Introducing Calculus: Can Change Occur at an Instant?

Defining Limits and Using Limit Notation

Estimating Limit Values from Graphs

Estimating Limit Values from Tables

Determining Limits Using Algebraic Properties of Limits

Determining Limits Using Algebraic Manipulation

Selecting Procedures for Determining Limits

Determining Limits Using the Squeeze Theorem

Connecting Multiple Representations of Limits

Exploring Types of Discontinuities

Defining Continuity at a Point

Confirming Continuity over an Interval

Removing Discontinuities

Connecting Infinite Limits and Vertical Asymptotes

Connecting Limits at Infinity and Horizontal Asymptotes

Working with the Intermediate Value Theorem (IVT)

Lesson

Diagnostic

after diagnostic

Key forms For every problem in this unit

Average rate on [a, b]

(f(b) − f(a)) / (b − a)

Instantaneous rate at a

limit of that quotient as b → a

Two-sided limit exists

left limit = right limit (and both exist)

Indeterminate forms

0/0, ∞/∞, 0·∞, ∞−∞: do more work

Nonzero over zero: k/0, k ≠ 0

unbounded. Check the SIGN from each side: one side can be +∞ and the other −∞, and the two-sided limit is infinite only when they AGREE

Squeeze Theorem

g ≤ f ≤ h near a, and g, h → L, so f → L

Standard trig limit

sin(u)/u → 1 as u → 0

Continuous at a

f(a) defined, limit exists, and the two are equal

Removable (hole)

sides agree, value missing or different

Jump

sides finite and unequal

Infinite

a side unbounded: vertical asymptote

Vertical asymptote

factor SURVIVES in the reduced denominator

Hole

factor CANCELS against the numerator

Horizontal asymptote (rational f)

compare DEGREES of the two polynomials: top < bottom → 0; equal → ratio of leading coefficients; top > bottom → NO HORIZONTAL one (divide out: top exactly one higher gives a SLANT asymptote, higher still gives a polynomial one)

IVT

f continuous on [a, b], N between f(a) and f(b): f(c) = N for some c

Unit 1 tools

Challenge bank

1 / 60

60 open-ended problems.

Read the question, work it out, then flip the card to compare your reasoning to the worked solution. Mark each card so you can return to the ones that still bite.

Question

Tap card to reveal explanation

Worked solution

Tap card to return to question

Cumulative assessment

Test the unit.

Twenty mixed items drawn from across all 16 topics, with guaranteed misconception-code coverage. Identifies which misconceptions still bite when you cannot see which topic the question came from.