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*}
Topic: Limits
 One Sided Limits
Teacher Preparation and Notes
 Students should already have been introduced to onesided 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 TI89 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
 Student Worksheet: OneSided 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*}
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*}
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}(21)\right) & = 3 \sin \left(\frac{\pi}{2}(24)\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.