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.
 
|-
 
|-
−
| | '''Slide Number 5'''
+
| | Slide Number 5
  
−
'''Limits'''
+
Limits
  
 
[[Image:]][[Image:]]
 
[[Image:]][[Image:]]
  
−
[[Image:]]
+
| | Let us understand the concept of '''limits'''.
−
| | 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'''.
 
Imagine yourself sliding along the curve or line towards a given value of '''x'''.
Line 63: Line 62:
 
|-
 
|-
 
| | '''Slide Number 6'''
 
| | '''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 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'''
 
'''Limit of a rational polynomial function'''
  
  
−
Let us find '''lim_(x→2) (3x<sup>2</sup> – x -10)/(x<sup>2</sup> – 4)'''
+
Let us find '''lim <u>(3x<sup>2</sup> – x -10)</u>'''
  
 +
'''x→2 (x<sup>2</sup> – 4)'''
 
| |
 
| |
  
Line 199: Line 76:
 
| | 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<sup>2</sup>-x-10)/(x<sup>2</sup>-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.
 +
  
 
In the '''input bar''', first type the '''numerator'''.
 
In the '''input bar''', first type the '''numerator'''.
Line 282: Line 161:
  
 
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 292: Line 174:
 
|-
 
|-
 
| | 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'''.
Line 299: Line 184:
  
 
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.
  
  
Line 306: Line 193:
 
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 12'''
+
| | '''Slide Number 7'''
  
−
'''Limit of a rational polynomial function'''
+
'''Limits of discontinuous functions'''
 +
 
 +
 
 +
[[Image:]]
 +
 
 +
'''lim h(x) = ?'''
 +
 
 +
'''x→c'''
 +
 
 +
'''lim h(x) = L4; lim h(x) = L3'''
  
−
'''lim_(x→2) (3x<sup>2</sup> – x -10)/(x<sup>2</sup> – 4) = 2.75'''
+
'''x→c- x→c+'''
 +
Thus, '''lim h(x)''' Does Not Exist ('''DNE''')
  
 +
          '''x→c'''
 
| |
 
| |
  
−
Thus, the '''limit''' of this '''rational polynomial function''' as '''x''' tends to 2 is 2.75.
+
 
 +
 
 +
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 13'''
+
| | '''Slide Number 8'''
  
 
'''Limit of a discontinuous function'''
 
'''Limit of a discontinuous function'''
  
  
−
Let us find '''lim_(x→0) f(x) = 2x+3, x ≤ 0'''
+
Let us find '''lim f(x) = 2x+3, x ≤ 0'''
  
−
................................  ='''3(x+1), x > 0'''
+
                '''x→0'''    '''3(x+1), x > 0'''
  
−
and '''lim_(x→1) f(x) = 2x+3, x ≤ 0'''
+
and '''lim f(x) = 2x+3, x ≤ 0'''
  
−
...........................= '''3(x+1), x > 0'''
+
    '''x→1'''  '''3(x+1), x > 0'''
 
| |
 
| |
  
Line 348: Line 265:
 
|  | 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 362: Line 280:
 
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.
 +
 +
  
 
|-
 
|-
Line 376: Line 300:
 
|-
 
|-
 
| | 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.
Line 394: Line 322:
 
| | 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 and press '''Enter'''.
+
 
 +
 
 +
| | 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'''
  
  
Line 402: Line 339:
  
 
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).'''
 +
 +
 +
 +
 +
 +
 +
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.
 
| | 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.
+
| | 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'''.
Line 431: Line 393:
 
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.
 
|-
 
|-
−
| | 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'''.
 
| | 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'''.
 
|-
 
|-
−
| | 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.
 
| | 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.
 
|-
 
|-
−
| | 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''' 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.
 
|-
 
|-
−
| | '''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.
 
| | Let us summarize.
 
|-
 
|-
−
| | '''Slide Number 15'''
+
| | '''Slide Number 9'''
  
 
'''Summary'''
 
'''Summary'''
Line 477: Line 425:
  
 
Look at continuity of '''functions'''
 
Look at continuity of '''functions'''
 +
  
 
|-
 
|-
−
| | '''Slide Number 16'''
+
| | '''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 '''lim_(x→0) sin 4x/sin 2x'''
+
Evaluate '''lim <u>sin4x'''</u>
  
 +
'''x→0'''              '''sin 2x'''
 
| | '''As an Assignment''':
 
| | '''As an Assignment''':
  
Line 493: Line 444:
 
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 17'''
+
| | '''Slide Number 11'''
  
 
'''About Spoken Tutorial project'''
 
'''About Spoken Tutorial project'''
Line 500: Line 451:
 
Please download and watch it.
 
Please download and watch it.
 
|-
 
|-
−
| | '''Slide Number 18'''
+
| | '''Slide Number 12'''
  
 
'''Spoken Tutorial workshops'''
 
'''Spoken Tutorial workshops'''
Line 511: Line 462:
 
For more details, please write to us.
 
For more details, please write to us.
 
|-
 
|-
−
| | '''Slide Number 19'''
+
| | '''Slide Number 13'''
  
 
'''Forum for specific questions:'''
 
'''Forum for specific questions:'''
Line 526: Line 477:
 
| | Please post your timed queries on this forum.
 
| | Please post your timed queries on this forum.
 
|-
 
|-
−
| | '''Slide Number 20'''
+
| | '''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


Let us find lim (3x2 – x -10)

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.


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.
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 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


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


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

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.


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.

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.


Its graph is seen in Graphics view.

Point to the break between the blue and red functions for f(x)=3(x+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. 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


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

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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.

Contributors and Content Editors

Madhurig, Vidhya

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