<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation
Our Terms of Use (click here to view) and Privacy Policy (click here to view) have changed. By continuing to use this site, you are agreeing to our new Terms of Use and Privacy Policy.

2.3: One Sided Limits

Difficulty Level: At Grade Created by: CK-12

This activty is intended to supplement Calculus, Chapter 1, Lesson 6.

ID: 10994

Time required: 15 minutes

Activity Overview

Students will graph piecewise functions and evaluate both the left hand limit and the right hand limit of the function as x approaches a given number, c. The Trace feature will be used to graphically estimate the one sided limit. Students will also use a table of values of each function to numerically verify that the values of the function to left and right of c are approaching the same number.

Topic: Limits

  • One Sided Limits

Teacher Preparation and Notes

  • Students should already have been introduced to one-sided limits.
  • Students should know that a limit exists if and only if the left hand limit and the right hand limit are equal.
  • Upgrade the TI-89 Titanium to OS Version 3.10 so that “Discontinuity Detection” can be utilized. On a graph, press F1 > Format to turn Discontinuity Detection ON.

Associated Materials

The student worksheet gives key press instructions to set up the window so that their graphs look like the following.

Problem 1

Before changing the value of a, students will graphically estimate the limit of y1(x) as x approaches 1 from the left and the right. Students will also use the table to numerically estimate the value of a that will ensure that the limit of y1(x) as x approaches one exists.

Solutions

limx1y1(x)limx1+y1(x)a1;5;1

Problem 2

Problem 1 is repeated for a different function. Before changing the value of a, students will graphically estimate the limit of y2(x) as x approaches 1 from the left and the right. Students will also use the table to numerically estimate the value of a that will ensure that the limit of y2(x) as x approaches one exists. Here the algebraic calculations for the left and right hand limits are to be shown.

Solutions

limx1 y2(x)limx1 y2(x)limx1+ y2(x)So 1+23;limx1+ y2(x)5;a3=1+2=3=a(12)=a=a12; a=3

Problem 3

Problems 1 and 2 are repeated for a different function. Before changing the value of a, students will graphically estimate the limit of y3(x) as x approaches 2 from the left and the right. Students will also use the table to numerically estimate the value of a that will ensure that the limit of y3(x) as x approaches two exists. Here the algebraic calculations for the left and right hand limits are to be shown.

Students should view the table near x=2 instead of 1.

Solutions

limx2 f1(x)2sin(π2(21))2;limx2+ f1(x)5;a2;=3sin(π2(24))+a;a=2

Extension – Continuity

Students are introduced to the concept of continuity and are asked to prove the functions in Problems 2 and 3 are continuous. They are also instructed how to use CAS to algebraically solve for a that makes the limit exist.

Solutions

Students must show that all three conditions are met in order to satisfy the criteria for continuity.

My Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Show More

Image Attributions

Show Hide Details
Files can only be attached to the latest version of section
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original
 
TI.MAT.ENG.TE.1.Calculus.2.3
Here