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# Limits and Continuity

## Objective

State and evaluate limits algebraically.

## Criteria for Success

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1. Identify the appropriate equations to use from a piecewise function to evaluate the left-hand limit, right-hand limit, and limit of the boundaries of a piecewise function.
2. Distinguish finding the value of a function at an ${{x-}}$value from finding the limit as you approach that ${{x-}}$value.
3. Verify algebraic reasoning graphically.

## 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

LO2.1A, LO2.1B (approaching)

## Anchor Problems

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### Problem 1

Below is a piecewise function.

${f(x)\left\{\begin{matrix}-x-2, \space \space -2\leq x <0 \\3x-2, \space \space \space 0\leq x <1 \\ x-3, \space \space \space 1\leq x \leq 4 \end{matrix}\right.}$

Calculate the following:

${\lim_{x\rightarrow 0}f(x)=}$

${\lim_{x\rightarrow 1}f(x)=}$

How can you tell if the function is continuous without graphing?

### Problem 2

Use ${f(x)={{x^2+6x+8}\over{x+2}}}$ to evaluate:

 a.    ${\lim_{x\rightarrow -2} f(x)=}$ b.  ${\lim_{x\rightarrow 2} f(x)=}$ c.  ${f(-2)=}$ d.  ${f(2)=}$

## 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.

• Include problems where there is a graph provided, but students need to show their work algebraically.
• Include problems where a piecewise function is given algebraically
• Include specific problems with right hand and left hand limits.
• Review skills from the rest of the unit, and ensure that students are writing some piecewise functions from graphs using parent functions other than linear.

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### Problem 1

Use ${f(x)={{x^2-7x+6}\over{x-6}}}$  to evaluate:

 a.  ${\lim_{x\rightarrow6}f(x)=}$ b.  ${\lim_{x\rightarrow-1}f(x)=}$ c.  ${\lim_{x\rightarrow0}f(x)=}$ d.  ${f(6)=}$ e.  ${f(-1)=}$ f.  ${f(0)=}$

### Problem 2

Use $g(x)=\left\{\begin{matrix} x+2 & x<-1 \\ x^2 & -1 \leq x <2\\ -2 x + 8 & 2 < x \leq 4 \end{matrix}\right.$  to evaluate:

 a.  ${\lim_{x\rightarrow-1} g(x)=}$ b.  ${\lim_{x\rightarrow2}g(x)=}$ c.  ${g(2)=}$ d.  ${g(4)=}$ e.  ${g(-1)=}$ f.  ${\lim_{x\rightarrow-\infty}g(x)=}$

Is this function $g$ continuous over the interval ${[0, 4]}$? How do you know?