# 2.3: One-Sided Limits

**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 screen by pressing .

2. Turn the functions off or clear them; press **Clear Functions**.

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

3. Turn Discontinuity Detection on. Press **Format** to find the option for Discontinuity Detection.

4. Set the window, using , to the settings shown at the right.

5. Back on the screen enter three piecewise functions.

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

For , type **when**

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

For , type **when**

For , type **when**

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

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

7. For Problems 1 and 2 below, use to have table settings of and , to numerically examine the left and right-hand limits. Be sure to press **ENTER** to save changes before pressing to view the table.

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

## Problem 1

Try other values for in the graph of to find what makes exist. On the screen, press **ENTER** when is highlighted. Press and then backspace to try different values for . Graph it to see if appear continuous.

## Problem 2

Try other values for in the graph of to find what makes exist.

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

## Problem 3

Try other values for in the graph of to find what makes exist.

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

## Extension – Continuity

A function is continuous at if:

- exists
- exists, and

Use CAS to algebraically solve for that makes

(a) exist

(b) exist

Then prove each function is continuous.

Key press help:

- Begin by pressing
**HOME**. Clean Up the screen by pressing . Choose**NewProb**and press**ENTER**to put this on the command line and**ENTER**to execute the command. - Type
**ENTER**. The Define command is under the menu. Type**Define**, 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 limit (. On the command line enter
**limit****ENTER**. - Now, press
**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**).