Applications-of-GeoGebra/C3/Limits-and-Continuity-of-Functions/English - Revision history

Madhurig at 17:12, 11 December 2018

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Vidhya at 06:45, 17 October 2018

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Vidhya: Created page with " {|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this '''tutorial''' on '''Limits and Continuity of Function..."

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Created page with " {|border=1 | | '''Visual Cue''' | | '''Narration''' |- | | '''Slide Number 1''' '''Title Slide''' | | Welcome to this '''tutorial''' on '''Limits and Continuity of Function..."

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{|border=1
| | '''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:]]

[[Image:]]
| | Let us understand the concept of '''limits''' by looking at three graphs '''A, B''' and '''C'''.

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'''

'''Left hand and right hand limits'''

[[Image:]]

'''lim_(x→b) f(x) = ?'''

'''lim_(x→ b-) f(x) = L1; lim_(x→b+) f(x) = L1 = f(b)'''

| |

In graph '''A''', let us find the '''limit''' of '''f of x''' as '''x''' approaches or tends to '''b'''.

'''f of x''' is a continuous line.

The '''left hand limit''' of '''f of x''' as '''x''' tends to '''b''' is '''L1'''.

And the '''right hand limit''' of '''f of x''' as '''x''' tends to '''b''' is also '''L1'''.

Thus, the '''limit''' of '''f of x''' as '''x''' approaches '''b''' is '''L1'''.

It is the same as evaluating '''f of x''' at '''x equals b''', that is, '''f of b.'''
|-
| | '''Slide Number 7'''

'''Left hand and right hand limits'''

[[Image:]]

'''lim_(x→b1) g(x) =?'''

'''lim_(x→b1-) g(x) = lim_(x→b1+) g(x) = L2''''

But '''g(b1)''' does not exist ('''DNE''')

'''lim_(x→b) g(x) = g(b) = L2; lim_(x→a) g(x) = g(a) = L1'''

| |

What is the '''limit''' of '''g of x''' as '''x''' tends to '''b1'''?

In graph '''A''', note that '''g of x''' has an open circle at '''b1 comma L2 prime'''.

This means that '''g of x''' does not exist at this point.

Let us find the '''limit''' of '''g of x''' as '''x''' approaches '''b1'''.

The '''left hand''' and '''right hand limits''' are '''L2'''' as '''x''' approaches '''b1'''.

But '''g of x''' itself does not exist at '''x equals b1'''.

However, '''g of x''' can be evaluated at '''x equals b''' and '''x equals a'''.

And these values are the same as the '''limits''' of '''g of x''' as '''x''' approaches '''b''' and '''a'''.
|-
| | '''Slide Number 8'''

'''Limits of discontinuous functions'''

[[Image:]]

'''lim_(x→c) h(x) = ?'''

'''lim_(x→c-) h(x) = L4; lim_(x→c+) h(x) = L3'''

Thus, '''lim_(x→c) h(x)''' does not exist ('''DNE''')

| |

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 9'''

'''Criteria for continuous functions'''

Note that for a '''function f(x)''' to be called continuous,

'''f(a)''' should be defined

'''lim_(x→ a) f(x)''' exists

'''lim_(x→ a) f(x) = f(a)'''

| |

Note that for a '''function f of x''' to be called continuous,

'''f of a''' should be defined

limit of '''f of x''' as '''x''' tends to '''a''' exists

limit of '''f of x''' as '''x''' tends to '''a''' is '''f of a'''
|-
| | '''Slide Number 10'''

'''Limits at infinity'''

[[Image:]]

'''lim_(x→∞) i(x) = ? lim_(x→-∞) i(x) = ?'''

'''lim_(x→∞) i(x) = 2; lim_(x→-∞) i(x) = 1'''

| |

In graph '''C''', '''i of x''' has two parts.

The first part is the upper right one.

Both arms extend towards '''infinity''' ('''∞''').

The second part is the lower left one.

Both arms extend towards '''negative infinity''' ('''-∞''').

What are the limits of '''i of x''' as '''x''' tends to '''infinity''' and '''minus infinity'''?

The limit of '''i of x''' as '''x''' approaches '''infinity''' is 2.

And the limit of '''i of x''' as '''x''' approaches '''negative infinity''' is 1.
|-
| | '''Slide Number 11'''

'''Limit of a rational polynomial function'''

Let us find '''lim_(x→2) (3x2 – x -10)/(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 x2-x-10)/(x2-4)''' 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'''.

In parentheses, 3 space '''x caret''' 2 minus '''x''' minus 10 followed by division slash'''

Now, type the '''denominator'''.

In parentheses, '''x caret''' 2 minus 4

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.

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'''.

Point to the spreadsheet.

| | Note that a question mark appears in '''cell B5''' corresponding to '''x equals 2'''.

This is because the '''function''' is undefined at this value.

The reason for this is that the denominator of the '''function''' becomes 0.

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.
|-
| | '''Slide Number 12'''

'''Limit of a rational polynomial function'''

'''lim_(x→2) (3x2 – x -10)/(x2 – 4) = 2.75'''

| |

Thus, the '''limit''' of this '''rational polynomial function''' as '''x''' tends to 2 is 2.75.
|-
| | '''Slide Number 13'''

'''Limit of a discontinuous function'''

Let us find '''lim_(x→0) f(x) = 2x+3, x ≤ 0'''

................................ ='''3(x+1), x > 0'''

and '''lim_(x→1) f(x) = 2x+3, x ≤ 0'''

...........................= '''3(x+1), x > 0'''
| |

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.

'''a''' equals '''Function''' with capital F and in square brackets '''2x plus 3''' comma minus 5 comma 0'''

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.
|-
| | 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.
| | 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.
|-
| | 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 and press '''Enter'''.

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.

For this piece of the '''function''', '''x''' is greater than 0 but not equal to 0.
|-
| | 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 is seen 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 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.
|-
| | Point to the break between the blue and red '''functions''' for '''f(x)=3(x+1).'''
| | 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'''.
|-
| | Point to the blue '''function'''.

Point to intersection of '''f(x)''' and '''y-axis''' at '''(0,3)'''.

| | 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.
|-
| | Point to the red '''function'''.
| | The red '''function''' has to be considered for '''x''' greater than 0.

When '''x''' equals 1, the value of '''f of x''' is 6.
|-
| | '''Slide Number 14'''

'''Limit of a discontinuous function'''

'''lim_(x→0) f(x) = 2x+3, x ≤ 0 }=3'''

..........................= '''3(x+1), x > 0'''

and '''lim_(x→1) f(x) = 2x+3, x ≤ 0 }=6'''

.........................= '''3(x+1), x > 0'''
| |

Thus, for this '''discontinuous function''', '''f of x''' is 3 when '''x''' is 0.

When '''x''' is 1, '''f of x''' is 6.
|-
| |
| | Let us summarize.
|-
| | '''Slide Number 15'''

'''Summary'''
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:

Understand limits of '''functions'''

Look at continuity of '''functions'''

|-
| | '''Slide Number 16'''

'''Assignment'''

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

Evaluate '''lim_(x→0) sin 4x/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 17'''

'''About Spoken Tutorial project'''
| | The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
|-
| | '''Slide Number 18'''

'''Spoken Tutorial workshops'''
| | The '''Spoken Tutorial Project''' team:

<nowiki>* conducts workshops using spoken tutorials and</nowiki>

<nowiki>* gives certificates on passing online tests.</nowiki>

For more details, please write to us.
|-
| | '''Slide Number 19'''

'''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 20'''

'''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.
|-
|}

@@ Line 344: / Line 344: @@
 |-
 ||Point to the red function.
-||The red '''function''' has to be considered for '''x''' greater than 0.
+||The red '''function''' has to be considered for '''x''' less
 |-
 ||Point to the blue function.
-||The blue '''function''' has to be considered for '''x''' less than and equal to 0.
+||The blue '''function''' has to be considered for '''x''' greater than 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 red function.
-||The red '''function''' has to be considered for '''x''' greater than 0.
 When '''x''' equals 1, the value of '''f of x''' is 6.

@@ Line 43: / Line 43: @@
 For relevant '''tutorials''', please visit our website.
 |-
-||Slide Number 5
+||'''Slide Number 5'''
-Limits
+'''Limits'''
-||Limits.
+||'''Limits'''
 Let us understand the concept of '''limits'''.
@@ Line 121: / Line 121: @@
 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.
@@ Line 132: / Line 130: @@
 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.
@@ Line 169: / Line 166: @@
 ||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.
+|-
-Observe that as '''x''' tends to 2, '''y''' tends to 2.75.
+||Point to the spreadsheet.
@@ Line 198: / Line 195: @@
 Thus, '''lim h(x)''' Does Not Exist ('''DNE''')
-           '''x→c'''
+'''x→c'''
 ||In graph '''B''', '''h of x''' is a '''piecewise''' or '''discontinuous function'''.
@@ Line 225: / Line 222: @@
 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'''
-     '''x→1'''  '''3(x+1), x > 0'''
+'''x→1'''  '''3(x+1), x > 0'''
 ||Limit of a discontinuous function.
@@ Line 246: / Line 243: @@
 ||Type '''a=Function[2x+3,-5,0]''' in the '''input bar''' >> '''Enter'''
 || In the '''input bar''', type the following line.
@@ Line 267: / Line 262: @@
 |-
 ||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.
@@ Line 276: / Line 269: @@
 |-
 ||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.
 |-
-||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.
-||Similarly, click on '''Move Graphics View''' and place the '''cursor''' on the '''y-axis'''.
+||Similarly, place the '''cursor''' on the '''y-axis'''.
@@ Line 297: / Line 287: @@
 |-
 ||Type '''b=Function[3(x+1),0.01,5]''' in the '''input bar''' >> '''Enter'''
 ||In the '''input bar''', type the following command.
@@ Line 338: / Line 329: @@
 Click on '''Move Graphics View''' and drag the background
-||Under '''Move Graphics View''', click on '''Zoom In'''.
+||Under '''Move Graphics View''', click on '''Zoom In'''
+and click in '''Graphics''' view to magnify the graph.
-Now click on '''Move Graphics View''' and drag the background until you can see both graphs.
 |-
-||
+||click on '''Move Graphics View''' >>
-||Note that there is a break between the blue and red '''functions'''.
+Drag the background to see both graphs.
 This is because '''x''' is not 0 when '''f of x''' is '''3 times x plus 1'''.
 |-
 ||Point to the blue function.
 ||The blue '''function''' has to be considered for '''x''' less than and equal to 0.

@@ Line 154: / Line 154: @@
 But the limit of '''h of x''' as '''x''' approaches '''c,''' '''does not exist''' ('''DNE''').
 |-
 | | '''Slide Number 10'''
@@ Line 222: / Line 199: @@
 | | I have already opened the '''GeoGebra''' interface.
 |-
 | | To type the '''caret symbol''', hold the '''Shift''' key down and press 6.
@@ Line 232: / Line 208: @@
 In the '''input bar''', first type the '''numerator'''.
 Now, type the '''denominator'''.
 Press '''Enter'''.
@@ Line 327: / Line 299: @@
 This is because the '''function''' is undefined at this value.
@@ Line 378: / Line 348: @@
 |  | In the '''input bar''', type the following line.
@@ Line 429: / Line 398: @@
 Remember the space denotes multiplication.
 This chooses the '''domain''' of '''x''' from 5 (for practical purposes) to 0.01.