# Limits and Continuity

## Objective

Sketch functions given limits and continuity requirements.

## Criteria for Success

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1. Use specifications about limits to sketch functions.
2. Identify and use continuity requirements to sketch functions.
3. Use limit notation appropriately for right-hand, left-hand, and actual limits.

## Tips for Teachers

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This lesson is aligned to the Learning Objectives and Essential Knowledge described in the College Board's AP Calculus AB and AP Calculus BC Course and Exam Description:

LO1.1A(b): EK1.1A1, EK1.1A2, EK1.1A3

LO1.1B: EK1.1B1

LO1.2A: EK1.2A1

Note: This lesson will also serve as the review for this unit.

## Anchor Problems

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### Problem 1

For the function ${f(x)}$, given below, determine the following:

Let ${f(x)}=\begin{Bmatrix} x^2-5 & x\leq 3 \\ x+2 & x>3 \end{Bmatrix}$

1. The left-hand limit at ${{{{x=3}}}}$
2. The right-hand limit at ${{{{x=3}}}}$
3. The limit at ${{{{x=3}}}}$
4. The value of the function at ${{{{x=3}}}}$

### Problem 2

Draw a picture of a function with the following properties:

${\lim_{x\rightarrow a^+}f(x)\neq \lim_{x\rightarrow a^-}f(x)}$

${\lim_{x\rightarrow a^+}f(x) = f(a)}$

## Problem Set

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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

• Use this problem set as an opportunity review all concepts from the unit.
• Include error analysis by giving a list of requirements and a graph with errors. Ask students to adjust the requirements to meet the graph or adjust the graph to meet the requirements.
• Give requirements that indicate different types of point discontinuities and jump discontinuities.

Draw the picture of a function where ${f(c)=\lim_{x\rightarrow c^+}f(x)}$, but the limit at ${x=c}$ does not exist.