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.
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