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2.3: One Sided Limits

Difficulty Level: At Grade Created by: CK-12
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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 \begin{align*}x\end{align*}x approaches a given number, \begin{align*}c\end{align*}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 \begin{align*}c\end{align*}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 \begin{align*}F1\end{align*}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 \begin{align*}a\end{align*}a, students will graphically estimate the limit of \begin{align*}y1(x)\end{align*}y1(x) as \begin{align*}x\end{align*}x approaches 1 from the left and the right. Students will also use the table to numerically estimate the value of \begin{align*}a\end{align*}a that will ensure that the limit of \begin{align*}y1(x)\end{align*}y1(x) as \begin{align*}x\end{align*}x approaches one exists.

Solutions

\begin{align*}\lim_{x \to 1^-} y1(x) & \approx 1; \\ \lim_{x \to 1^+}y1(x) & \approx 5; \\ a & \approx 1\end{align*}limx1y1(x)limx1+y1(x)a1;5;1

Problem 2

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

Solutions

\begin{align*}\lim_{x \to 1^-} \ y2(x) & \approx 3; \lim_{x \to 1^+} \ y2(x) \approx 5; a \approx 3 \\ \lim_{x \to 1^-} \ y2(x) & = 1 + 2 = 3 \\ \lim_{x \to 1^+} \ y2(x) & = a \cdot(1^2) = a \\ \text{So} \ 1 + 2 & = a \cdot 1^2; \quad \ a = 3 \end{align*}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 \begin{align*}a\end{align*}a, students will graphically estimate the limit of \begin{align*}y3(x)\end{align*}y3(x) as \begin{align*}x\end{align*}x approaches 2 from the left and the right. Students will also use the table to numerically estimate the value of \begin{align*}a\end{align*}a that will ensure that the limit of \begin{align*}y3(x)\end{align*}y3(x) as \begin{align*}x\end{align*}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 \begin{align*}x = 2\end{align*}x=2 instead of 1.

Solutions

\begin{align*}\lim_{x \to 2^-} \ f1(x) & \approx 2; \lim_{x \to 2^+} \ f1(x) \approx 5; a \approx 2; \\ 2\sin \left(\frac{\pi}{2}(2-1)\right) & = 3 \sin \left(\frac{\pi}{2}(2-4)\right) + a; a = 2\end{align*}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 \begin{align*}a\end{align*}a that makes the limit exist.

Solutions

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

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