Limits and Continuity

Lesson 7

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?

Guiding Questions

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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.  $${f(2)=}$$

Guiding Questions

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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. 

Target Task

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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?