Difference between revisions of "Applications-of-GeoGebra/C3/Limits-and-Continuity-of-Functions/English"
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For relevant '''tutorials''', please visit our website. | For relevant '''tutorials''', please visit our website. | ||
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− | | | | + | | | Slide Number 5 |
− | + | Limits | |
[[Image:]][[Image:]] | [[Image:]][[Image:]] | ||
− | + | | | 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'''. | ||
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| | '''Slide Number 6''' | | | '''Slide Number 6''' | ||
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'''Limit of a rational polynomial function''' | '''Limit of a rational polynomial function''' | ||
− | Let us find ''' | + | Let us find '''lim <u>(3x<sup>2</sup> – x -10)</u>''' |
+ | '''x→2 (x<sup>2</sup> – 4)''' | ||
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| | 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 | + | 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. | ||
+ | |||
In the '''input bar''', first type the '''numerator'''. | In the '''input bar''', first type the '''numerator'''. | ||
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Place the '''cursor''' at the bottom right corner of the '''cell'''. | Place the '''cursor''' at the bottom right corner of the '''cell'''. | ||
+ | |||
+ | |||
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Drag the '''cursor''' to highlight cells until '''B10'''. | Drag the '''cursor''' to highlight cells until '''B10'''. | ||
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| | 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. | 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'''. | ||
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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. | ||
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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. | ||
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− | | | '''Slide Number | + | | | '''Slide Number 7''' |
− | ''' | + | '''Limits of discontinuous functions''' |
+ | |||
+ | |||
+ | [[Image:]] | ||
+ | |||
+ | '''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''' | ||
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− | + | ||
+ | |||
+ | 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,''' '''does not exist''' ('''DNE'''). | ||
+ | |||
|- | |- | ||
− | | | '''Slide Number | + | | | '''Slide Number 8''' |
'''Limit of a discontinuous function''' | '''Limit of a discontinuous function''' | ||
− | Let us find ''' | + | Let us find '''lim f(x) = 2x+3, x ≤ 0''' |
− | + | '''x→0''' '''3(x+1), x > 0''' | |
− | and ''' | + | and '''lim f(x) = 2x+3, x ≤ 0''' |
− | + | '''x→1''' '''3(x+1), x > 0''' | |
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| | 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''' | ||
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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. | ||
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+ | |||
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| | 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'''. | | | Similarly, 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, click on '''Move Graphics View''' and 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. | ||
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| | 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. | ||
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Remember the space denotes multiplication. | Remember the space denotes multiplication. | ||
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+ | '''b''' equals '''Function''' with capital F | ||
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+ | In square brackets, type 3 space '''x''' plus 1 in parentheses comma 0.01 comma 5''' | ||
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For this piece of the '''function''', '''x''' is greater than 0 but not equal to 0. | For this piece of the '''function''', '''x''' is greater than 0 but not equal to 0. | ||
+ | |||
+ | Press '''Enter'''. | ||
|- | |- | ||
| | Point to the equation '''b(x)=3(x+1) (0.01 ≤ x ≤ 5)''' in '''Algebra''' view. | | | 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 is seen 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).''' | ||
+ | |||
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+ | Point to the blue '''function'''. | ||
+ | |||
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+ | Point to intersection of '''f(x)''' and '''y-axis''' at '''(0,3)'''. | ||
+ | |||
+ | |||
+ | Point to the red '''function'''. | ||
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| | Double click on the equation '''a(x)=2x+3''' 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'''. | ||
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Now 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. | ||
|- | |- | ||
− | | | | + | | | |
| | Note that there is a break between the blue and red '''functions'''. | | | Note that there is a break between the blue and red '''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'''. | ||
|- | |- | ||
− | | | | + | | | |
− | + | ||
− | + | ||
− | + | ||
| | The blue '''function''' has to be considered for '''x''' less than and equal to 0. | | | The blue '''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. | | | The red '''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. | ||
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| | Let us summarize. | | | Let us summarize. | ||
|- | |- | ||
− | | | '''Slide Number | + | | | '''Slide Number 9''' |
'''Summary''' | '''Summary''' | ||
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Look at continuity of '''functions''' | Look at continuity of '''functions''' | ||
+ | |||
|- | |- | ||
− | | | '''Slide Number | + | | | '''Slide Number 10''' |
'''Assignment''' | '''Assignment''' | ||
+ | |||
Find the limit of '''(x<sup>3</sup>-2x<sup>2</sup>)/(x<sup>2</sup>-5x+6)''' as '''x''' tends to 2. | Find the limit of '''(x<sup>3</sup>-2x<sup>2</sup>)/(x<sup>2</sup>-5x+6)''' as '''x''' tends to 2. | ||
− | Evaluate ''' | + | Evaluate '''lim <u>sin4x'''</u> |
+ | '''x→0''' '''sin 2x''' | ||
| | '''As an Assignment''': | | | '''As an Assignment''': | ||
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Find the limit of this '''trigonometric function''' as '''x''' tends to 0. | Find the limit of this '''trigonometric function''' as '''x''' tends to 0. | ||
|- | |- | ||
− | | | '''Slide Number | + | | | '''Slide Number 11''' |
'''About Spoken Tutorial project''' | '''About Spoken Tutorial project''' | ||
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Please download and watch it. | Please download and watch it. | ||
|- | |- | ||
− | | | '''Slide Number | + | | | '''Slide Number 12''' |
'''Spoken Tutorial workshops''' | '''Spoken Tutorial workshops''' | ||
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For more details, please write to us. | For more details, please write to us. | ||
|- | |- | ||
− | | | '''Slide Number | + | | | '''Slide Number 13''' |
'''Forum for specific questions:''' | '''Forum for specific questions:''' | ||
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| | Please post your timed queries on this forum. | | | Please post your timed queries on this forum. | ||
|- | |- | ||
− | | | '''Slide Number | + | | | '''Slide Number 14''' |
'''Acknowledgement''' | '''Acknowledgement''' |
Revision as of 12:06, 6 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 [[Image:]][[Image:]] |
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
x→2 (x2 – 4) |
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.
Let us graph functions and look at their limits. |
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.
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. |
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 and 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.
|
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 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.
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 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.
The reason for this is that the denominator of the function becomes 0.
|
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, 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 |
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.
a equals Function with capital F and in square brackets 2x plus 3 comma minus 5 comma 0
Press Enter. |
Point to the equation a(x)=2x+3 (-5 ≤ x ≤ 0) in Algebra view.
|
The equation a of x equals 2x plus 3 where x varies from minus 5 to 0 appears in Algebra 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.
|
Click on Move Graphics View and place the cursor on the x-'axis.
|
Similarly, click on Move Graphics View tool and place cursor on y-axis.
|
Similarly, click on Move Graphics View and place the cursor on the y-axis.
|
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.
For this piece of the function, x is greater than 0 but not equal to 0. Press Enter. |
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.
|
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. |
Click on Object Properties. | Click on Object Properties. |
Click on Color tab and 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 to see both functions in Graphics view. | |
Under Move Graphics View, click on Zoom In.
Now click on Move Graphics View and drag the background until you can see both graphs. | |
Note that there is a break between the blue and red functions.
This is because x is not 0 when f of x is 3 times x plus 1. | |
The blue 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. | |
The red 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 |
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. |