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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 $Y=$ screen by pressing $[\blacklozenge]$ $F1$.

2. Turn the functions off or clear them; press $F1 >$ Clear Functions.

Note: You can turn functions off by un-checking them using $F4$.

3. Turn Discontinuity Detection on. Press $F1 >$ Format to find the option for Discontinuity Detection.

4. Set the window, using $[\blacklozenge]$ $F2$, to the settings shown at the right.

5. Back on the $Y=$ screen enter three piecewise functions.

At $y1$ press ENTER. Find when(in the CATALOG quickly by pressing CATALOG [ . ]. This shows the notation: when(condition, true, false)

For $y1$, type when$(x<1,1,a)|a=5$

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

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

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

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

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

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

6. Graph one function at a time by using $F4$ to have only one function checked at a time.

On a graph screen examine both sides of where the discontinuity exists using $F3$ Trace.

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

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

## Problem 1

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

$\lim_{x \to 1^-}y1(x) \approx \underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

$\lim_{x \to 1^+}y1(x) \approx \underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

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

$a=\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

## Problem 2

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

$\lim_{x \to 1^-}y2(x) \approx \underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

$\lim_{x \to 1^+}y2(x)\approx \underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

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

$a=\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Show calculations of the left hand limit and the right hand limit to verify that your value for $a$ makes the limit exist.

## Problem 3

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

$\lim_{x \to 2^-} y3(x)\approx \underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

$\lim_{x \to 2^+}y3(x)\approx \underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

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

$a=\underline{\;\;\;\;\;\;\;\;\;\;\;\;}$

Show calculations of the left hand limit and the right hand limit to verify that your value for $a$ makes the limit exist.

## Extension – Continuity

A function is continuous at $x=c$ if:

• $f(c)$ exists
• $\lim_{x \to c}f(x)$ exists, and
• $\lim_{x \to c}f(x)=f(c)$

Use CAS to algebraically solve for $a$ that makes

(a) $\lim_{x \to 1}y2(x)$ exist

(b) $\lim_{x \to 2}y3(x)$ exist

Then prove each function is continuous.

Key press help:

• Begin by pressing HOME. Clean Up the screen by pressing $2^{nd}$ $[F1]$. Choose NewProb and press ENTER to put this on the command line and ENTER to execute the command.
• Type $y2(x)$ ENTER. The Define command is under the $F4$ menu. Type Define $f(x)=$, 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 $F3 >$ limit(. On the command line enter limit$(f(x),x,1,1)$ ENTER.
• Now, press $F2$ 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$(f(x),x,1,1) = \text{limit}\ (f(x) ,x, 1,-1),a$).

Feb 23, 2012

Nov 04, 2014