2.3: One Sided Limits
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*} approaches a given number, \begin{align*}c\end{align*}. 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*} 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*} > Format to turn Discontinuity Detection ON.
Associated Materials
- Student Worksheet: One-Sided Limits http://www.ck12.org/flexr/chapter/9726, scroll down to the third activity.
- Discontinuity Detection
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*}, students will graphically estimate the limit of \begin{align*}y1(x)\end{align*} as \begin{align*}x\end{align*} 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*} that will ensure that the limit of \begin{align*}y1(x)\end{align*} as \begin{align*}x\end{align*} 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*}
Problem 2
Problem 1 is repeated for a different function. Before changing the value of \begin{align*}a\end{align*}, students will graphically estimate the limit of \begin{align*}y2(x)\end{align*} as \begin{align*}x\end{align*} 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*} that will ensure that the limit of \begin{align*}y2(x)\end{align*} as \begin{align*}x\end{align*} 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*}
Problem 3
Problems 1 and 2 are repeated for a different function. Before changing the value of \begin{align*}a\end{align*}, students will graphically estimate the limit of \begin{align*}y3(x)\end{align*} as \begin{align*}x\end{align*} 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*} that will ensure that the limit of \begin{align*}y3(x)\end{align*} as \begin{align*}x\end{align*} 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*} 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*}
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*} that makes the limit exist.
Solutions
Students must show that all three conditions are met in order to satisfy the criteria for continuity.
Image Attributions
To add resources, you must be the owner of the section. Click Customize to make your own copy.