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# 2.3: One-Sided Limits

Difficulty Level: At Grade Created by: CK-12

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

## Set up – graphing piecewise functions that show discontinuity

1. After turning on your device, go to the \begin{align*}Y=\end{align*} screen by pressing \begin{align*}[\blacklozenge]\end{align*} \begin{align*}F1\end{align*}.

2. Turn the functions off or clear them; press \begin{align*}F1 >\end{align*} Clear Functions.

Note: You can turn functions off by un-checking them using \begin{align*}F4\end{align*}.

3. Turn Discontinuity Detection on. Press \begin{align*}F1 >\end{align*} Format to find the option for Discontinuity Detection.

4. Set the window, using \begin{align*}[\blacklozenge]\end{align*} \begin{align*}F2\end{align*}, to the settings shown at the right.

5. Back on the \begin{align*}Y=\end{align*} screen enter three piecewise functions.

At \begin{align*}y1\end{align*} press ENTER. Find when(in the CATALOG quickly by pressing CATALOG [ . ]. This shows the notation: when(condition, true, false)

For \begin{align*}y1\end{align*}, type when\begin{align*}(x<1,1,a)|a=5\end{align*}

The “such that” bar key ([ | ]) is to the left of the [7] key.

\begin{align*}y1(x)= \begin{cases} 1,x<1 \\ \qquad \quad \ |a=5 \\ a,x \ge 1 \end{cases}\end{align*}

For \begin{align*}y2\end{align*}, type when\begin{align*}(x<1,x+2,a^*x^{\land}2)|a=5\end{align*}

\begin{align*}y2(x)= \begin{cases} x+2, x < 1 \\ \qquad \qquad \quad |a=5 \\ a \cdot x^2, x \ge 1 \end{cases}\end{align*}

For \begin{align*}y3\end{align*}, type when \begin{align*}(x < 2,2 \sin \left((x - 1) \frac {\pi}{2} \right),a + 3 \sin \left((x-4)\frac {\pi}{2} \right)|a=5\end{align*}

\begin{align*}y3(x)= \begin{cases} 2 \sin \left ((x-1) \frac{\pi}{2} \right ) ,x<2 \\ \qquad \qquad \qquad \qquad \qquad |a=5 \\ a + 3 \sin \left ((x-4) \frac {\pi}{2} \right ),x \ge 12 \end{cases}\end{align*}

6. Graph one function at a time by using \begin{align*}F4\end{align*} to have only one function checked at a time.

On a graph screen examine both sides of where the discontinuity exists using \begin{align*}F3\end{align*} Trace.

7. For Problems 1 and 2 below, use \begin{align*}[\blacklozenge]\end{align*} \begin{align*}F4\end{align*} to have table settings of \begin{align*}\text{tblStart} = 0.98\end{align*} and \begin{align*}\triangle \text{tbl}= 0.01\end{align*}, to numerically examine the left and right-hand limits. Be sure to press ENTER to save changes before pressing \begin{align*}[\blacklozenge]\end{align*} \begin{align*}F5\end{align*} to view the table.

For Problems 1, 2, and 3 estimate the limits graphically and numerically using trace and table.

## Problem 1

\begin{align*}y1(x) = \begin{cases} 1,x<1 \\ \qquad \qquad |a=5 \\ a,x\ge1 \end{cases}\end{align*}

\begin{align*}\lim_{x \to 1^-}y1(x) \approx \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

\begin{align*}\lim_{x \to 1^+}y1(x) \approx \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

Try other values for \begin{align*}a\end{align*} in the graph of \begin{align*}y1(x)\end{align*} to find what \begin{align*}a\end{align*} makes \begin{align*}\lim_{x \to 1}y1(x)\end{align*} exist. On the \begin{align*}Y=\end{align*} screen, press ENTER when \begin{align*}y1\end{align*} is highlighted. Press \begin{align*} \blacktriangleright \end{align*} and then backspace \begin{align*}\leftarrow\end{align*} to try different values for \begin{align*}a\end{align*}. Graph it to see if appear continuous.

\begin{align*}a=\underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

## Problem 2

\begin{align*}y2(x)= \begin{cases} x+2, x<1 \\ \qquad \qquad \quad|a=5 \\ a \cdot x^2,x\ge1 \end{cases}\end{align*}

\begin{align*}\lim_{x \to 1^-}y2(x) \approx \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

\begin{align*}\lim_{x \to 1^+}y2(x)\approx \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

Try other values for \begin{align*}a\end{align*} in the graph of \begin{align*}y2(x)\end{align*} to find what \begin{align*}a\end{align*} makes \begin{align*}\lim_{x \to 1}y2(x)\end{align*} exist.

\begin{align*}a=\underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

Show calculations of the left hand limit and the right hand limit to verify that your value for \begin{align*}a\end{align*} makes the limit exist.

## Problem 3

\begin{align*}y3(x)= \begin{cases} 2\sin \left((x-1)\frac{\pi}{2} \right),x<2 \\ \qquad \qquad \qquad \qquad \qquad |a=5 \\ a+3\sin \left((x-4)\frac{\pi}{2} \right),x\ge2 \end{cases}\end{align*}

\begin{align*}\lim_{x \to 2^-} y3(x)\approx \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

\begin{align*}\lim_{x \to 2^+}y3(x)\approx \underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

Try other values for \begin{align*}a\end{align*} in the graph of \begin{align*}y3(x)\end{align*} to find what \begin{align*}a\end{align*} makes \begin{align*}\lim_{x \to 2} y3(x)\end{align*} exist.

\begin{align*}a=\underline{\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

Show calculations of the left hand limit and the right hand limit to verify that your value for \begin{align*}a\end{align*} makes the limit exist.

## Extension – Continuity

A function is continuous at \begin{align*}x=c\end{align*} if:

• \begin{align*}f(c)\end{align*} exists
• \begin{align*}\lim_{x \to c}f(x)\end{align*} exists, and
• \begin{align*}\lim_{x \to c}f(x)=f(c)\end{align*}

Use CAS to algebraically solve for \begin{align*}a\end{align*} that makes

(a) \begin{align*}\lim_{x \to 1}y2(x)\end{align*} exist

(b) \begin{align*}\lim_{x \to 2}y3(x)\end{align*} exist

Then prove each function is continuous.

Key press help:

• Begin by pressing HOME. Clean Up the screen by pressing \begin{align*}2^{nd}\end{align*} \begin{align*}[F1]\end{align*}. Choose NewProb and press ENTER to put this on the command line and ENTER to execute the command.
• Type \begin{align*}y2(x)\end{align*} ENTER. The Define command is under the \begin{align*}F4\end{align*} menu. Type Define \begin{align*}f(x)=\end{align*}, then up arrow to highlight the output from the previous line. Press ENTER on the highlighted piecewise function to copy it down to the command line.

• To solve a right sided limit, press \begin{align*}F3 > \end{align*} limit(. On the command line enter limit\begin{align*}(f(x),x,1,1)\end{align*} ENTER.
• Now, press \begin{align*}F2\end{align*} ENTER to select solve( Then up arrow to select the input from the previous line, press ENTER. Next type [ -- ]. Up arrow to the input again and press ENTER. This time put a negative (-) in front of the last 1. Finally type [ , ] ALPHA [ -- ] and close the parentheses. This method will enable you to quickly enter solve(limit\begin{align*}(f(x),x,1,1) = \text{limit}\ (f(x) ,x, 1,-1),a\end{align*}).

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