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Limits and Continuity

Lesson 7


State and evaluate limits algebraically.

Criteria for Success


  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


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


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


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


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?