Unit 1 · Core unit

Limits and Continuity

The vocabulary the rest of calculus is written in. A limit describes approach rather than arrival, which is why it can exist where the function does not.

What a strong answer looks like

A strong answer says which form the substitution produced before doing anything. Indeterminate and undefined are different diagnoses with different next steps.

Practice this unit Course overview

Chapter introduction

Understand the idea before memorizing the rule

The vocabulary the rest of calculus is written in. A limit describes approach rather than arrival, which is why it can exist where the function does not.

In Calculus Honors, success means moving among words, symbols, tables, and graphs while preserving the meaning of each quantity. Use the lessons below as connected notes: explain the concept, carry out the procedure, test the result, and interpret it in context.

Detailed study notes

Core lessons, reasoning, and common errors

Lesson 1

Evaluating a Limit

Concept explanation

Substitution is the first move. A number is the answer; a nonzero over zero signals an infinite limit; zero over zero means the expression can still be simplified.

How to apply it

Substitute first, then factor, rationalise, or use a known limit depending on which form appeared.

Common mistake

Reading 0/0 as undefined and stopping, when it is the one form that means keep going.

Guided study move

Sort five limits by the form substitution produces before evaluating any of them.

Lesson 2

One-Sided Behavior and Continuity

Concept explanation

A two-sided limit exists only when both sides agree. Continuity needs three things at once: the value exists, the limit exists, and they match.

How to apply it

Check the branches separately for a piecewise function, then compare them against the stated value.

Common mistake

Assuming a defined value implies continuity, or that a zero denominator always means an asymptote.

Guided study move

Classify each discontinuity of a rational function as removable, infinite, or jump.

Lesson 3

End Behavior and the Intermediate Value Theorem

Concept explanation

Limits at infinity describe end behavior and give horizontal asymptotes. The Intermediate Value Theorem turns continuity into a guarantee that a value is attained.

How to apply it

Compare degrees for a rational limit at infinity; check continuity and a sign change before invoking the theorem.

Common mistake

Believing a graph cannot cross a horizontal asymptote, or that the theorem locates a unique root.

Guided study move

Show a root exists on an interval, then say why the theorem does not tell you how many.

Formula and visual reference

Connect the symbols to the picture

Chapter synthesis

Ideas to connect

Naming the form before choosing a method
Treating one-sided limits separately
Stating the hypotheses before applying a theorem

Modeling lab

Transfer the chapter to a new setting

A piecewise function models a fee that changes at a threshold. Decide whether the charge is continuous there, and say what the one-sided limits mean for a customer at the boundary.

Mastery checklist

• Evaluate a limit that requires factoring.
• Decide whether a two-sided limit exists from one-sided values.
• Classify a discontinuity from a factored expression.
• Find a horizontal asymptote by comparing degrees.

Check yourself

• Why can a limit exist where the function value does not?
• What does the Intermediate Value Theorem guarantee, and what does it not?

Modeling drill

A piecewise function models a fee that changes at a threshold. Decide whether the charge is continuous there, and say what the one-sided limits mean for a customer at the boundary.

LimitOne-sided limitIndeterminate formContinuityRemovable discontinuityHorizontal asymptoteIntermediate Value Theorem

Unit 2: The Derivative and Differentiation Rules