Difference between revisions of "Applications-of-GeoGebra/C3/Limits-and-Continuity-of-Functions/English"

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{|border=1
 
{|border=1
−
| | '''Visual Cue'''
+
||'''Visual Cue'''
−
| | '''Narration'''
+
||'''Narration'''
  
 
|-
 
|-
−
| | '''Slide Number 1'''
+
||'''Slide Number 1'''
  
 
'''Title Slide'''
 
'''Title Slide'''
−
| | Welcome to this '''tutorial''' on '''Limits and Continuity of Functions'''.
+
||Welcome to this tutorial on '''Limits and Continuity of Functions'''.
 
|-
 
|-
−
| | '''Slide Number 2'''
+
||'''Slide Number 2'''
  
 
'''Learning Objectives'''
 
'''Learning Objectives'''
−
| | In this '''tutorial''', we will learn how to use '''GeoGebra''' to:
+
||In this '''tutorial''', we will learn how to use '''GeoGebra''' to:
  
 
Understand '''limits''' of '''functions'''
 
Understand '''limits''' of '''functions'''
Line 19: Line 19:
 
Look at continuity of '''functions'''
 
Look at continuity of '''functions'''
 
|-
 
|-
−
| | '''Slide Number 3'''
+
||'''Slide Number 3'''
  
 
'''System Requirement'''
 
'''System Requirement'''
−
| | Here I am using:
+
||Here I am using:
  
 
'''Ubuntu Linux''' OS version 16.04
 
'''Ubuntu Linux''' OS version 16.04
Line 28: Line 28:
 
'''GeoGebra''' 5.0.481.0-d
 
'''GeoGebra''' 5.0.481.0-d
 
|-
 
|-
−
| | '''Slide Number 4'''
+
||'''Slide Number 4'''
  
 
'''Pre-requisites'''
 
'''Pre-requisites'''
  
 
'''www.spoken-tutorial.org'''
 
'''www.spoken-tutorial.org'''
−
| | To follow this '''tutorial''', you should be familiar with:
+
||To follow this tutorial, you should be familiar with:
  
 
'''GeoGebra''' interface
 
'''GeoGebra''' interface
Line 43: Line 43:
 
For relevant '''tutorials''', please visit our website.
 
For relevant '''tutorials''', please visit our website.
 
|-
 
|-
−
| | Slide Number 5
+
||'''Slide Number 5'''
  
−
Limits
+
'''Limits'''
  
−
[[Image:]][[Image:]]
+
||'''Limits'''
  
−
| | Let us understand the concept of '''limits'''.
+
Let us understand the concept of '''limits'''.
  
 
Imagine yourself sliding along the curve or line towards a given value of '''x'''.
 
Imagine yourself sliding along the curve or line towards a given value of '''x'''.
Line 61: Line 61:
 
The right side gives the '''right hand limit'''.
 
The right side gives the '''right hand limit'''.
 
|-
 
|-
−
| | '''Slide Number 6'''
+
||'''Slide Number 6'''
  
 
'''Limit of a rational polynomial function'''
 
'''Limit of a rational polynomial function'''
−
 
  
 
Let us find '''lim <u>(3x<sup>2</sup> – x -10)</u>'''
 
Let us find '''lim <u>(3x<sup>2</sup> – x -10)</u>'''
  
 
'''x→2 (x<sup>2</sup> – 4)'''
 
'''x→2 (x<sup>2</sup> – 4)'''
−
| |
+
||'''Limit of a rational polynomial function'''
  
 
Let us find the '''limit''' of this '''rational polynomial function''' as '''x''' tends to 2.
 
Let us find the '''limit''' of this '''rational polynomial function''' as '''x''' tends to 2.
 
|-
 
|-
−
| | Show the '''GeoGebra''' window.
+
||Show the '''GeoGebra''' window.
−
| | I have already opened the '''GeoGebra''' interface.
+
||I have already opened the '''GeoGebra''' interface.
  
−
Let us graph functions and look at their limits.
 
 
|-
 
|-
−
| | To type the '''caret symbol''', hold the '''Shift''' key down and press 6.
+
||To type the '''caret symbol''', hold the '''Shift''' key down and press 6.
  
 
Type '''(3 x^2-x-10)/(x^2-4)''' in the '''input bar''' >> '''Enter'''
 
Type '''(3 x^2-x-10)/(x^2-4)''' in the '''input bar''' >> '''Enter'''
−
| | To type the '''caret symbol''', hold the '''Shift''' key down and press 6.
+
||To type the '''caret symbol''', hold the '''Shift''' key down and press 6.
  
 
Note that spaces denote multiplication.
 
Note that spaces denote multiplication.
Line 92: Line 90:
 
Press '''Enter'''.
 
Press '''Enter'''.
 
|-
 
|-
−
| | Point to the equation in '''Algebra''' view and its graph in '''Graphics''' view.
+
||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.
+
||The equation appears in '''Algebra''' view and its graph in '''Graphics''' view.
 
|-
 
|-
−
| | Click on '''Move Graphics View''' tool.
+
||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 in and drag '''Graphics''' view to see the graph.
−
| | Click on '''Move Graphics View'''.
+
||Click on '''Move Graphics View'''.
  
 
Click in and drag '''Graphics''' view to see the graph.
 
Click in and drag '''Graphics''' view to see the graph.
 
|-
 
|-
−
| | Point to the graph in '''Graphics''' view.
+
||Point to the graph in '''Graphics''' view.
−
| | As '''x''' approaches 2, the '''function''' approaches some value close to 3.
+
||As '''x''' approaches 2, the '''function''' approaches some value close to 3.
 
|-
 
|-
−
| | Click on '''View''' tool and select '''Spreadsheet'''.
+
||Click on '''View''' tool >> select '''Spreadsheet'''.
−
| | Click on '''View''' and select '''Spreadsheet'''.
+
||Click on '''View''' and select '''Spreadsheet'''.
 
|-
 
|-
−
| | Point to the spreadsheet on the right side of the '''Graphics''' view.
+
||Point to the spreadsheet on the right side of the '''Graphics''' view.
−
| | This opens a 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''' tool and click on '''Rounding''' and choose '''5 decimal places'''.
−
| | Click on '''Options''' 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.
−
 
+
−
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.
 
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.
+
||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.
 
We will choose values of '''x''' less than but close to 2.
−
 
  
 
Remember to press '''Enter''' to go to the next '''cell'''.
 
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.
 
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.
−
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.
 
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 '''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)^2-A1-10)/((A1)^2-4)''' >> '''Enter'''.
+
||In '''cell B1''' (that is, '''column B, cell 1'''), type '''(3(A1)^2-A1-10)/((A1)^2-4)''' >> '''Enter'''.
−
| | In '''cell B1''' (that is, '''column B, cell 1'''), type the following ratio of values.
+
||In '''cell B1''' (that is, '''column B, cell 1'''), type the following ratio of values.
  
 
First, the numerator in parentheses
 
First, the numerator in parentheses
Line 150: Line 144:
 
'''A1''' in parentheses '''caret''' 2 minus 4  and press '''Enter'''.
 
'''A1''' in parentheses '''caret''' 2 minus 4  and press '''Enter'''.
 
|-
 
|-
−
| | Click on '''cell B1''' to highlight it.
+
||Click on '''cell B1''' to highlight it.
  
 
Place the '''cursor''' at the bottom right corner of the '''cell'''.
 
Place the '''cursor''' at the bottom right corner of the '''cell'''.
Line 158: Line 152:
  
 
Point to '''y''' values in '''column B''' and to the '''x''' values in '''column A'''.
 
Point to '''y''' values in '''column B''' and to the '''x''' values in '''column A'''.
−
| | Click on '''cell B1''' to highlight it.
+
||Click on '''cell B1''' to highlight it.
  
 
Place the '''cursor''' at the bottom right corner of the '''cell'''.
 
Place the '''cursor''' at the bottom right corner of the '''cell'''.
−
 
−
 
−
 
  
 
Drag the '''cursor''' to highlight cells until '''B10'''.
 
Drag the '''cursor''' to highlight cells until '''B10'''.
Line 170: Line 161:
 
This fills in '''y''' values corresponding to the '''x''' values in '''column A'''.
 
This fills in '''y''' values corresponding to the '''x''' values in '''column A'''.
 
|-
 
|-
−
| | Drag and increase column width.
+
||Drag and increase column width.
−
| | Drag and increase column width.
+
||Drag and increase column width.
 
|-
 
|-
−
| | Point to the '''question mark''' in '''cell B5''' corresponding to '''x=2'''.
+
||Point to the '''question mark''' in '''cell B5''' corresponding to '''x=2'''.
−
 
+
−
 
+
−
Point to the spreadsheet.
+
−
 
+
−
 
+
−
 
+
−
| | Note that a question mark appears in '''cell B5''' corresponding to '''x equals 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.
 
This is because the '''function''' is undefined at this value.
−
 
+
|-
−
The reason for this is that the denominator of the '''function''' becomes 0.
+
||Point to the spreadsheet.
−
 
+
||Observe that as '''x''' tends to 2, '''y''' tends to 2.75.
−
 
+
−
Observe that as '''x''' tends to 2, '''y''' tends to 2.75.
+
  
  
 
Hence, as '''x''' tends to 2, the limit of the '''function''' tends to 2.75.
 
Hence, as '''x''' tends to 2, the limit of the '''function''' tends to 2.75.
 
|-
 
|-
−
| | '''Slide Number 7'''
+
||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'''
 
'''Limits of discontinuous functions'''
−
 
−
 
−
[[Image:]]
 
  
 
'''lim h(x) = ?'''
 
'''lim h(x) = ?'''
Line 209: Line 195:
 
Thus, '''lim h(x)''' Does Not Exist ('''DNE''')
 
Thus, '''lim h(x)''' Does Not Exist ('''DNE''')
  
−
          '''x→c'''
+
'''x→c'''
−
| |
+
||In graph '''B''', '''h of x''' is a '''piecewise''' or '''discontinuous function'''.
−
 
+
−
 
+
−
 
+
−
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'''.
 
We want to find the '''limit''' of '''h of x''' as '''x''' approaches '''c'''.
Line 230: Line 212:
 
The '''left''' and '''right hand limits''' exist.
 
The '''left''' and '''right hand limits''' exist.
  
−
But the limit of '''h of x''' as '''x''' approaches '''c,''' '''does not exist''' ('''DNE''').
+
But the limit of '''h of x''' as '''x''' approaches '''c, itself does not exist''' ('''DNE''').
  
 
|-
 
|-
−
| | '''Slide Number 8'''
+
||'''Slide Number 8'''
  
 
'''Limit of a discontinuous function'''
 
'''Limit of a discontinuous function'''
Line 240: Line 222:
 
Let us find '''lim f(x) = 2x+3, x ≤ 0'''
 
Let us find '''lim f(x) = 2x+3, x ≤ 0'''
  
−
                '''x→0'''    '''3(x+1), x > 0'''
+
'''x→0'''    '''3(x+1), x > 0'''
  
 
and '''lim f(x) = 2x+3, x ≤ 0'''
 
and '''lim f(x) = 2x+3, x ≤ 0'''
  
−
    '''x→1'''  '''3(x+1), 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'''.
 
Let us find limits of a '''piecewise''' or '''discontinuous function f of x'''.
Line 256: Line 238:
 
We want to find the limits when '''x''' tends to 0 and 1.
 
We want to find the limits when '''x''' tends to 0 and 1.
 
|-
 
|-
−
| | Open a new '''GeoGebra''' window.
+
||Open a new '''GeoGebra''' window.
−
| | Let us open a new '''GeoGebra''' window.
+
||Let us open a new '''GeoGebra''' window.
 
|-
 
|-
−
| | Type '''a=Function[2x+3,-5,0]''' in the '''input bar''' >> '''Enter'''
+
||Type '''a=Function[2x+3,-5,0]''' in the '''input bar''' >> '''Enter'''
−
 
+
|| In the '''input bar''', type the following line.
−
 
+
−
 
+
−
| | In the '''input bar''', type the following line.
+
−
 
+
−
'''a''' equals '''Function''' with capital F and in square brackets '''2x plus 3''' comma minus 5 comma 0'''
+
  
  
Line 272: Line 249:
 
Press '''Enter'''.
 
Press '''Enter'''.
 
|-
 
|-
−
| | Point to the equation '''a(x)=2x+3 (-5 ≤ x ≤ 0)''' in '''Algebra''' view.
+
||Point to the equation '''a(x)=2x+3 (-5 ≤ x ≤ 0)''' in '''Algebra''' view.
−
 
+
  
 
Drag the boundary to see it properly.
 
Drag the boundary to see it properly.
−
 
  
 
Point to its graph in '''Graphics''' view.
 
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.
+
||The equation '''a of x equals 2x plus 3''' where '''x''' varies from minus 5 to 0 appears in '''Algebra''' view.
  
  
 
Drag the boundary to see it properly.
 
Drag the boundary to see it properly.
−
 
  
 
Its graph is seen in '''Graphics''' 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.
−
 
+
|| 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''' 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 '''x-axis'''.
+
||Click on '''Move Graphics View''' tool, place '''cursor''' on '''x-axis'''.
−
 
+
  
 
When an arrow appears along the '''axis''', drag the '''x-axis''' to zoom in or out.
 
When an arrow appears along the '''axis''', drag the '''x-axis''' to zoom in or out.
−
| | Click on '''Move Graphics View''' and place the '''cursor''' on the '''x-'axis'''.
+
||Click on '''Move Graphics View''' and place the '''cursor''' on the '''x-'axis'''.
−
 
+
  
 
When an arrow appears along the '''axis''', drag the '''x-axis''' to zoom in or out.
 
When an arrow appears along the '''axis''', drag the '''x-axis''' to zoom in or out.
 
|-
 
|-
−
| | Similarly, click on '''Move Graphics View''' tool and place '''cursor''' on '''y-axis'''.
+
||Click on '''Move Graphics View''' tool and place '''cursor''' on '''y-axis'''.
−
 
+
  
 
When an arrow appears along the '''axis''', drag the '''y-axis''' to zoom in or out.
 
When an arrow appears along the '''axis''', drag the '''y-axis''' to zoom in or out.
−
| | Similarly, click on '''Move Graphics View''' and place the '''cursor''' on the '''y-axis'''.
+
||Similarly, place the '''cursor''' on the '''y-axis'''.
  
  
 
When an arrow appears along the '''axis''', drag the '''y-axis''' to zoom in or out.
 
When an arrow appears along the '''axis''', drag the '''y-axis''' to zoom in or out.
 
|-
 
|-
−
| | Click in and drag the background to see the graph properly.
+
||Click in and drag the background to see the graph properly.
−
| | 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'''
+
||Type '''b=Function[3(x+1),0.01,5]''' in the '''input bar''' >> '''Enter'''
  
−
 
+
||In the '''input bar''', type the following command.
−
 
+
−
| | In the '''input bar''', type the following command.
+
  
  
 
Remember the space denotes multiplication.
 
Remember the space denotes multiplication.
−
 
−
 
−
'''b''' equals '''Function''' with capital F
 
−
 
−
 
−
In square brackets, type 3 space '''x''' plus 1 in parentheses comma 0.01 comma 5'''
 
−
 
  
 
This chooses the '''domain''' of '''x''' from 5 (for practical purposes) to 0.01.
 
This chooses the '''domain''' of '''x''' from 5 (for practical purposes) to 0.01.
Line 342: Line 299:
 
Press '''Enter'''.   
 
Press '''Enter'''.   
 
|-
 
|-
−
| | Point to the equation '''b(x)=3(x+1) (0.01 ≤ x ≤ 5)''' in '''Algebra''' view.
+
||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.
 
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.
  
  
−
| | The equation '''b of x equals 3 times x plus 1''' where '''x''' varies from 0.01 to 5 appears in '''Algebra''' view.
+
Its graph appears in '''Graphics''' view.
−
 
+
−
 
+
−
Its graph is seen in '''Graphics''' view.
+
 
|-
 
|-
−
| | Point to the break between the blue and red '''functions''' for '''f(x)=3(x+1).'''
+
||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.
−
 
+
−
 
+
−
 
+
−
 
+
−
 
+
−
Point to the blue '''function'''.
+
−
 
+
−
 
+
−
Point to intersection of '''f(x)''' and '''y-axis''' at '''(0,3)'''.
+
−
 
+
−
 
+
−
Point to the red '''function'''.
+
−
 
+
−
 
+
−
 
+
−
| |
+
−
 
+
 
|-
 
|-
−
| | Double click on the equation '''a(x)=2x+3''' in '''Algebra''' view.
+
||Click on '''Object Properties'''.
−
| | 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 '''Object Properties'''.
+
||Click on the '''Color''' tab and select blue.
 
|-
 
|-
−
| | Click on '''Color''' tab and select blue.
+
||Close the '''Preferences''' dialog box.
−
| | Click on the '''Color''' tab and select blue.
+
||Close the '''Preferences''' dialog box.
 
|-
 
|-
−
| | Close the '''Preferences''' dialog box.
+
||Click in and drag the background.
−
| | Close the '''Preferences''' dialog box.
+
||Click in and drag the background to see both '''functions''' in '''Graphics''' view.
 
|-
 
|-
−
| |
+
||Under '''Move Graphics View''', click on '''Zoom In'''.
−
| | Click in and drag the background to see both '''functions''' in '''Graphics''' view.
+
 
 +
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''' >>
−
| | Under '''Move Graphics View''', click on '''Zoom In'''.
+
Drag the background to see both graphs.
−
 
+
||Again click on '''Move Graphics View''' and drag the background until you can see both graphs.
−
Now click on '''Move Graphics View''' and drag the background until you can see both graphs.
+
 
|-
 
|-
−
| |
+
||Point to the break between the blue and
−
| | Note that there is a break between the blue and red '''functions'''.
+
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'''.
 
This is because '''x''' is not 0 when '''f of x''' is '''3 times x plus 1'''.
 
|-
 
|-
−
| |
+
||Point to the red function.
−
| | The blue '''function''' has to be considered for '''x''' less than and equal to 0.
+
||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 '''y-axis''' at 0 comma 3.
 
When '''x''' tends to 0, '''f of x''' is 3 as the '''function''' intersects the '''y-axis''' at 0 comma 3.
 +
 +
 
|-
 
|-
−
| |
+
 
−
| | The red '''function''' has to be considered for '''x''' greater than 0.
+
 
 +
||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.
 
When '''x''' equals 1, the value of '''f of x''' is 6.
 
|-
 
|-
−
| |
+
||
−
| |
+
||Let us summarize.
−
|-
+
−
| |
+
−
| |
+
−
|-
+
−
| |
+
−
| | Let us summarize.
+
 
|-
 
|-
−
| | '''Slide Number 9'''
+
||'''Slide Number 9'''
  
 
'''Summary'''
 
'''Summary'''
−
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:
+
||In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:
  
 
Understand limits of '''functions'''
 
Understand limits of '''functions'''
Line 428: Line 372:
  
 
|-
 
|-
−
| | '''Slide Number 10'''
+
||'''Slide Number 10'''
  
 
'''Assignment'''
 
'''Assignment'''
Line 438: Line 382:
  
 
'''x→0'''              '''sin 2x'''
 
'''x→0'''              '''sin 2x'''
−
| | '''As an Assignment''':
+
||'''As an Assignment''':
  
 
Find the limit of this '''rational polynomial function''' as '''x''' tends to 2.
 
Find the limit of this '''rational polynomial function''' as '''x''' tends to 2.
Line 444: Line 388:
 
Find the limit of this '''trigonometric function''' as '''x''' tends to 0.
 
Find the limit of this '''trigonometric function''' as '''x''' tends to 0.
 
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'''About Spoken Tutorial project'''
 
'''About Spoken Tutorial project'''
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| | The video at the following link summarizes the '''Spoken Tutorial''' project.
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||The video at the following link summarizes the '''Spoken Tutorial''' project.
  
 
Please download and watch it.
 
Please download and watch it.
 
|-
 
|-
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| | '''Slide Number 12'''
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||'''Slide Number 12'''
  
 
'''Spoken Tutorial workshops'''
 
'''Spoken Tutorial workshops'''
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| | The '''Spoken Tutorial Project''' team:
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||The '''Spoken Tutorial Project''' team:
  
 
<nowiki>* conducts workshops using spoken tutorials and</nowiki>
 
<nowiki>* conducts workshops using spoken tutorials and</nowiki>
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For more details, please write to us.
 
For more details, please write to us.
 
|-
 
|-
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'''Forum for specific questions:'''
 
'''Forum for specific questions:'''
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Someone from our team will answer them
 
Someone from our team will answer them
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| | Please post your timed queries on this forum.
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||Please post your timed queries on this forum.
 
|-
 
|-
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| | '''Slide Number 14'''
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||'''Slide Number 14'''
  
 
'''Acknowledgement'''
 
'''Acknowledgement'''
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| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.
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||'''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.
  
 
More information on this mission is available at this link.
 
More information on this mission is available at this link.
 
|-
 
|-
 
| |
 
| |
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| | This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.
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||This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.
  
 
Thank you for joining.
 
Thank you for joining.
 
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Latest revision as of 22:42, 11 December 2018

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 5.0.481.0-d

Slide Number 4

Pre-requisites

www.spoken-tutorial.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 (3x2 – x -10)

x→2 (x2 – 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^2-x-10)/(x^2-4) in the input bar >> Enter

To type the caret symbol, hold the Shift key down and press 6.

Note that spaces denote multiplication.


In the input bar, first type the numerator.

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)^2-A1-10)/((A1)^2-4) >> 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.


Drag the cursor to highlight cells until B10.

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.


This fills in y values corresponding to the x values in column A.

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.


Hence, as x tends to 2, the limit of the function 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


Let us find lim f(x) = 2x+3, x ≤ 0

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.


f of x is described by 2x plus 3 when x is 0 or less than 0.

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.


This chooses the domain of x from minus 5 (for practical purposes) to 0.

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.


Drag the boundary to see it properly.

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 x-axis.

When an arrow appears along the axis, drag the x-axis 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 x-axis to zoom in or out.

Click on Move Graphics View tool and place cursor on y-axis.

When an arrow appears along the axis, drag the y-axis to zoom in or out.

Similarly, place the cursor on the y-axis.


When an arrow appears along the axis, drag the y-axis to zoom in or out.

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.


Remember the space denotes multiplication.

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.


Its graph appears in Graphics 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 y-axis 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


Find the limit of (x3-2x2)/(x2-5x+6) as x tends to 2.

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

The video at the following link summarizes the Spoken Tutorial project.

Please download and watch it.

Slide Number 12

Spoken Tutorial workshops

The Spoken Tutorial Project team:

* conducts workshops using spoken tutorials and

* gives certificates on passing online tests.

For more details, please write to us.

Slide Number 13

Forum for specific questions:

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them

Please post your timed queries on this forum.
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.

Contributors and Content Editors

Madhurig, Vidhya

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