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

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Line 33: Line 33:
  
 
'''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 47: Line 47:
 
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 64: Line 64:
  
 
'''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.
Line 76: Line 75:
 
||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.
Line 94: Line 92:
 
||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.
 +
|-
 +
||Drag the boundary.
 +
||Drag the boundary to see both properly.
 
|-
 
|-
 
||Click on '''Move Graphics View''' tool.
 
||Click on '''Move Graphics View''' tool.
Line 105: Line 106:
 
||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'''.
 
|-
 
|-
Line 114: Line 115:
 
||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.
Line 129: Line 128:
 
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.
−
 
+
−
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.
 
||Let us find the '''right hand limit''' of this '''function''' as '''x''' tends to 2.
  
Line 161: Line 158:
  
 
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 174: Line 168:
 
|-
 
|-
 
||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'''.
−
 
  
 
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.
 
−
 
  
 
Observe that as '''x''' tends to 2, '''y''' tends to 2.75.
 
Observe that as '''x''' tends to 2, '''y''' tends to 2.75.
Line 192: Line 178:
  
 
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.
 +
|-
 +
||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'''
 
||'''Slide Number 7'''
  
 
'''Limits of discontinuous functions'''
 
'''Limits of discontinuous functions'''
−
 
−
 
−
[[Image:]]
 
  
 
'''lim h(x) = ?'''
 
'''lim h(x) = ?'''
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           '''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 215:
 
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''').
  
 
|-
 
|-
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     '''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 260: Line 245:
 
|-
 
|-
 
||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'''
 
'''a''' equals '''Function''' with capital F and in square brackets '''2x plus 3''' comma minus 5 comma 0'''
Line 273: Line 255:
 
|-
 
|-
 
||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.
Line 283: Line 263:
  
 
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.
Line 290: Line 269:
  
  
−
 
+
|| 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 321: Line 297:
 
|-
 
|-
 
||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 341: Line 307:
  
 
Press '''Enter'''.   
 
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 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 appears 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.
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||Click on '''Object Properties'''.
 
||Click on '''Object Properties'''.
 
|-
 
|-
−
||Click on '''Color''' tab and select blue.
+
||Click on '''Color''' tab >> select blue.
 
||Click on the '''Color''' tab and select blue.
 
||Click on the '''Color''' tab and select blue.
 
|-
 
|-
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||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.
 
||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'''.
 
||Under '''Move Graphics View''', click on '''Zoom In'''.
  
 
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'''.
 
|-
 
|-
−
| |
+
||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''' 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.
 
|-
 
|-
−
| |
+
||
−
| |
+
−
|-
+
−
| |
+
−
| |
+
−
|-
+
−
| |
+
 
||Let us summarize.
 
||Let us summarize.
 
|-
 
|-

Revision as of 18:15, 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.

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.

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.

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.

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.

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.

Point to the blue function. 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.

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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