ApplicationsofGeoGebra/C3/LimitsandContinuityofFunctions/English
Visual Cue  Narration 
Slide Number 1
Title Slide 
Welcome to this tutorial on Limits and Continuity of Functions. 
Slide Number 2
Learning Objectives 
In this tutorial, we will learn how to use GeoGebra to:
Understand limits of functions Look at continuity of functions 
Slide Number 3
System Requirement 
Here I am using:
Ubuntu Linux OS version 16.04 GeoGebra 5.0.481.0d 
Slide Number 4
Prerequisites www.spokentutorial.org 
To follow this tutorial, you should be familiar with:
GeoGebra interface Limits Elementary calculus For relevant tutorials, please visit our website. 
Slide Number 5
Limits 
Limits
Let us understand the concept of limits. Imagine yourself sliding along the curve or line towards a given value of x. The height at which you will be, is the corresponding y value of the function. Any value of x can be approached from two sides. The left side gives the left hand limit. The right side gives the right hand limit. 
Slide Number 6
Limit of a rational polynomial function Let us find lim (3x^{2} – x 10) x→2 (x^{2} – 4) 
Limit of a rational polynomial function
Let us find the limit of this rational polynomial function as x tends to 2. 
Show the GeoGebra window.  I have already opened the GeoGebra interface. 
To type the caret symbol, hold the Shift key down and press 6.
Type (3 x^2x10)/(x^24) in the input bar >> Enter 
To type the caret symbol, hold the Shift key down and press 6.
Note that spaces denote multiplication.
Now, type the denominator. Press Enter. 
Point to the equation in Algebra view and its graph in Graphics view.  The equation appears in Algebra view and its graph in Graphics view. 
Drag the boundary.  Drag the boundary to see both properly. 
Click on Move Graphics View tool.
Click in and drag Graphics view to see the graph. 
Click on Move Graphics View.
Click in and drag Graphics view to see the graph. 
Point to the graph in Graphics view.  As x approaches 2, the function approaches some value close to 3. 
Click on View tool >> select Spreadsheet.  Click on View and select Spreadsheet. 
Point to the spreadsheet on the right side of the Graphics view.  This opens a spreadsheet on the right side of the Graphics view. 
Click on Options tool and click on Rounding and choose 5 decimal places.  Click on Options and click on Rounding and choose 5 decimal places. 
Remember to press Enter to go to the next cell.
Type 1.91, 1.93, 1.96, 1.98 and 2 in column A from cells 1 to 5. 
Let us find the left hand limit of this function as x tends to 2.
We will choose values of x less than but close to 2. Remember to press Enter to go to the next cell. In column A in cells 1 to 5, type 1.91, 1.93, 1.96, 1.98 and 2. 
Type 2.01, 2.03, 2.05, 2.07 and 2.09 in column A from cells 6 to 10.  Let us find the right hand limit of this function as x tends to 2.
We will choose values of x greater than but close to 2. In column A from cells 6 to 10, type 2.01, 2.03, 2.05, 2.07 and 2.09. 
In cell B1 (that is, column B, cell 1), type (3(A1)^2A110)/((A1)^24) >> Enter.  In cell B1 (that is, column B, cell 1), type the following ratio of values.
First, the numerator in parentheses 3 A1 in parentheses caret 2 minus A1 minus 10 followed by division slash Now the denominator in parentheses A1 in parentheses caret 2 minus 4 and press Enter. 
Click on cell B1 to highlight it.
Place the cursor at the bottom right corner of the cell.
Point to y values in column B and to the x values in column A. 
Click on cell B1 to highlight it.
Place the cursor at the bottom right corner of the cell. Drag the cursor to highlight cells until B10.

Drag and increase column width.  Drag and increase column width. 
Point to the question mark in cell B5 corresponding to x=2.  Note that a question mark appears in cell B5 corresponding to x equals 2.
This is because the function is undefined at this value. 
Point to the spreadsheet.  Observe that as x tends to 2, y tends to 2.75.

Click in Graphics view and drag the background
to see this properly. 
Click in Graphics view and drag the background
to see this properly. 
Slide Number 7
Limits of discontinuous functions lim h(x) = ? x→c lim h(x) = L4; lim h(x) = L3 x→c x→c+ Thus, lim h(x) Does Not Exist (DNE) x→c 
In graph B, h of x is a piecewise or discontinuous function.
We want to find the limit of h of x as x approaches c. So let us look at the left and right hand limits. For the left hand limit, look at the lower limb where the limit is L4. For the right hand limit, look at the upper limb where limit of h of x is L3. But as x approaches c, the two limbs of h of x approach different values of y. These are L3 and L4. The left and right hand limits exist. But the limit of h of x as x approaches c, itself does not exist (DNE). 
Slide Number 8
Limit of a discontinuous function
x→0 3(x+1), x > 0 and lim f(x) = 2x+3, x ≤ 0 x→1 3(x+1), x > 0 
Limit of a discontinuous function.
Let us find limits of a piecewise or discontinuous function f of x.
But f of x is described by 3 times x plus 1 when x is greater than 0. We want to find the limits when x tends to 0 and 1. 
Open a new GeoGebra window.  Let us open a new GeoGebra window. 
Type a=Function[2x+3,5,0] in the input bar >> Enter  In the input bar, type the following line.
Press Enter. 
Point to the equation a(x)=2x+3 (5 ≤ x ≤ 0) in Algebra view.
Drag the boundary to see it properly. Point to its graph in Graphics view. 
The equation a of x equals 2x plus 3 where x varies from minus 5 to 0 appears in Algebra view.
Its graph is seen in Graphics view. 
Under Move Graphics View, click on Zoom Out and click in Graphics view.  Under Move Graphics View, click on Zoom Out and click in Graphics view. 
Click on Move Graphics View and drag the background to see the graph properly.  Click on Move Graphics View and drag the background to see the graph properly. 
Click on Move Graphics View tool, place cursor on xaxis.
When an arrow appears along the axis, drag the xaxis to zoom in or out. 
Click on Move Graphics View and place the cursor on the x'axis.
When an arrow appears along the axis, drag the xaxis to zoom in or out. 
Click on Move Graphics View tool and place cursor on yaxis.
When an arrow appears along the axis, drag the yaxis to zoom in or out. 
Similarly, place the cursor on the yaxis.

Click in and drag the background to see the graph properly.  Click in and drag the background to see the graph properly. 
Type b=Function[3(x+1),0.01,5] in the input bar >> Enter  In the input bar, type the following command.
This chooses the domain of x from 5 (for practical purposes) to 0.01. For this piece of the function, x is greater than 0 but not equal to 0. Press Enter. 
Drag the boundary to see the equation properly.  Drag the boundary to see the equation properly. 
Point to the equation b(x)=3(x+1) (0.01 ≤ x ≤ 5) in Algebra view.
Point to its graph in Graphics view. 
The equation b of x equals 3 times x plus 1 where x varies from 0.01 to 5 appears in Algebra view.

Double click on the equation a(x)=2x+3 in Algebra view.  In Algebra view, double click on the equation b of x equals 3 times x plus 1. 
Click on Object Properties.  Click on Object Properties. 
Click on Color tab >> select blue.  Click on the Color tab and select blue. 
Close the Preferences dialog box.  Close the Preferences dialog box. 
Click in and drag the background.  Click in and drag the background to see both functions in Graphics view. 
Under Move Graphics View, click on Zoom In.
Click on Move Graphics View and drag the background 
Under Move Graphics View, click on Zoom In
and click in Graphics view to magnify the graph. 
click on Move Graphics View >>
Drag the background to see both graphs. 
Again click on Move Graphics View and drag the background until you can see both graphs. 
Point to the break between the blue and
red functions. 
Continue to Zoom In and drag the background
until you see the gap between the functions. This is because x is not 0 when f of x is 3 times x plus 1. 
Point to the red function.  The red function has to be considered for x less
than and equal to 0. When x tends to 0, f of x is 3 as the function intersects the yaxis at 0 comma 3.

Point to the blue function.  The blue function has to be considered for x greater than 0.
When x equals 1, the value of f of x is 6. 
Let us summarize.  
Slide Number 9
Summary 
In this tutorial, we have learnt how to use GeoGebra to:
Understand limits of functions Look at continuity of functions

Slide Number 10
Assignment
Evaluate lim sin4x x→0 sin 2x 
As an Assignment:
Find the limit of this rational polynomial function as x tends to 2. Find the limit of this trigonometric function as x tends to 0. 
Slide Number 11
About Spoken Tutorial project 
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Please download and watch it. 
Slide Number 12
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Slide Number 13
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Slide Number 14
Acknowledgement 
Spoken Tutorial Project is funded by NMEICT, MHRD, Government of India.
More information on this mission is available at this link. 
This is Vidhya Iyer from IIT Bombay, signing off.
Thank you for joining. 