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

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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
 
{|border=1
−
| | '''Visual Cue'''
+
||'''Visual Cue'''
−
| | '''Narration'''
+
||'''Narration'''
  
 
|-
 
|-
−
| | '''Slide Number 1'''
+
||'''Slide Number 1'''
  
 
'''Title Slide'''
 
'''Title Slide'''
−
| | Welcome to this '''tutorial''' on '''Limits and Continuity of Functions'''.
+
||Welcome to this tutorial on '''Limits and Continuity of Functions'''.
 
|-
 
|-
−
| | '''Slide Number 2'''
+
||'''Slide Number 2'''
  
 
'''Learning Objectives'''
 
'''Learning Objectives'''
−
| | In this '''tutorial''', we will learn how to use '''GeoGebra''' to:
+
||In this '''tutorial''', we will learn how to use '''GeoGebra''' to:
  
 
Understand '''limits''' of '''functions'''
 
Understand '''limits''' of '''functions'''
Line 19: Line 19:
 
Look at continuity of '''functions'''
 
Look at continuity of '''functions'''
 
|-
 
|-
−
| | '''Slide Number 3'''
+
||'''Slide Number 3'''
  
 
'''System Requirement'''
 
'''System Requirement'''
−
| | Here I am using:
+
||Here I am using:
  
 
'''Ubuntu Linux''' OS version 16.04
 
'''Ubuntu Linux''' OS version 16.04
Line 28: Line 28:
 
'''GeoGebra''' 5.0.481.0-d
 
'''GeoGebra''' 5.0.481.0-d
 
|-
 
|-
−
| | '''Slide Number 4'''
+
||'''Slide Number 4'''
  
 
'''Pre-requisites'''
 
'''Pre-requisites'''
  
 
'''www.spoken-tutorial.org'''
 
'''www.spoken-tutorial.org'''
−
| | To follow this '''tutorial''', you should be familiar with:
+
||To follow this tutorial, you should be familiar with:
  
 
'''GeoGebra''' interface
 
'''GeoGebra''' interface
Line 43: Line 43:
 
For relevant '''tutorials''', please visit our website.
 
For relevant '''tutorials''', please visit our website.
 
|-
 
|-
−
| | '''Slide Number 5'''
+
||'''Slide Number 5'''
  
 
'''Limits'''
 
'''Limits'''
  
−
[[Image:]][[Image:]]
+
||'''Limits'''
  
−
[[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 62: Line 61:
 
The right side gives the '''right hand limit'''.
 
The right side gives the '''right hand limit'''.
 
|-
 
|-
−
| | '''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 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'''
 
'''Limit of a rational polynomial function'''
  
 +
Let us find '''lim <u>(3x<sup>2</sup> – x -10)</u>'''
  
−
Let us find '''lim_(x→2) (3x<sup>2</sup> – x -10)/(x<sup>2</sup> – 4)'''
+
'''x→2 (x<sup>2</sup> – 4)'''
−
 
+
||'''Limit of a rational polynomial function'''
−
| |
+
  
 
Let us find the '''limit''' of this '''rational polynomial function''' as '''x''' tends to 2.
 
Let us find the '''limit''' of this '''rational polynomial function''' as '''x''' tends to 2.
 
|-
 
|-
−
| | Show the '''GeoGebra''' window.
+
||Show the '''GeoGebra''' window.
−
| | I have already opened the '''GeoGebra''' interface.
+
||I have already opened the '''GeoGebra''' interface.
  
−
Let us graph functions and look at their limits.
 
 
|-
 
|-
−
| | To type the '''caret symbol''', hold the '''Shift''' key down and press 6.
+
||To type the '''caret symbol''', hold the '''Shift''' key down and press 6.
  
−
Type '''(3 x<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'''.
−
 
−
In parentheses, 3 space '''x caret''' 2 minus '''x''' minus 10 followed by division slash'''
 
  
 
Now, type the '''denominator'''.
 
Now, type the '''denominator'''.
−
 
−
In parentheses, '''x caret''' 2 minus 4
 
  
 
Press '''Enter'''.
 
Press '''Enter'''.
 
|-
 
|-
−
| | Point to the equation in '''Algebra''' view and its graph in '''Graphics''' view.
+
||Point to the equation in '''Algebra''' view and its graph in '''Graphics''' view.
−
| | The equation appears in '''Algebra''' view and its graph in '''Graphics''' view.
+
||The equation appears in '''Algebra''' view and its graph in '''Graphics''' view.
 +
|-
 +
||Drag the boundary.
 +
||Drag the boundary to see both properly.
 
|-
 
|-
−
| | Click on '''Move Graphics View''' tool.
+
||Click on '''Move Graphics View''' tool.
  
 
Click in and drag '''Graphics''' view to see the graph.
 
Click in and drag '''Graphics''' view to see the graph.
−
| | Click on '''Move Graphics View'''.
+
||Click on '''Move Graphics View'''.
  
 
Click in and drag '''Graphics''' view to see the graph.
 
Click in and drag '''Graphics''' view to see the graph.
 
|-
 
|-
−
| | Point to the graph in '''Graphics''' view.
+
||Point to the graph in '''Graphics''' view.
−
| | As '''x''' approaches 2, the '''function''' approaches some value close to 3.
+
||As '''x''' approaches 2, the '''function''' approaches some value close to 3.
 
|-
 
|-
−
| | Click on '''View''' tool and select '''Spreadsheet'''.
+
||Click on '''View''' tool >> select '''Spreadsheet'''.
−
| | Click on '''View''' and select '''Spreadsheet'''.
+
||Click on '''View''' and select '''Spreadsheet'''.
 
|-
 
|-
−
| | Point to the spreadsheet on the right side of the '''Graphics''' view.
+
||Point to the spreadsheet on the right side of the '''Graphics''' view.
−
| | This opens a spreadsheet on the right side of the '''Graphics''' view.
+
||This opens a spreadsheet on the right side of the '''Graphics''' view.
 
|-
 
|-
−
| | Click on '''Options''' tool and click on '''Rounding''' and choose '''5 decimal places'''.
+
||Click on '''Options''' tool and click on '''Rounding''' and choose '''5 decimal places'''.
−
| | Click on '''Options''' and click on '''Rounding''' and choose '''5 decimal places'''.
+
||Click on '''Options''' and click on '''Rounding''' and choose '''5 decimal places'''.
 
|-
 
|-
−
| |
+
||Remember to press '''Enter''' to go to the next cell.
−
 
+
−
Remember to press '''Enter''' to go to the next cell.
+
  
 
Type 1.91, 1.93, 1.96, 1.98 and 2 in '''column A''' from '''cells''' 1 to 5.
 
Type 1.91, 1.93, 1.96, 1.98 and 2 in '''column A''' from '''cells''' 1 to 5.
−
| | Let us find the '''left hand limit''' of this '''function''' as '''x''' tends to 2.
+
||Let us find the '''left hand limit''' of this '''function''' as '''x''' tends to 2.
  
 
We will choose values of '''x''' less than but close to 2.
 
We will choose values of '''x''' less than but close to 2.
−
 
  
 
Remember to press '''Enter''' to go to the next '''cell'''.
 
Remember to press '''Enter''' to go to the next '''cell'''.
−
 
  
 
In '''column A''' in '''cells''' 1 to 5, type 1.91, 1.93, 1.96, 1.98 and 2.
 
In '''column A''' in '''cells''' 1 to 5, type 1.91, 1.93, 1.96, 1.98 and 2.
 
|-
 
|-
−
| |
+
||Type 2.01, 2.03, 2.05, 2.07 and 2.09 in '''column A''' from '''cells''' 6 to 10.
−
 
+
||Let us find the '''right hand limit''' of this '''function''' as '''x''' tends to 2.
−
Type 2.01, 2.03, 2.05, 2.07 and 2.09 in '''column A''' from '''cells''' 6 to 10.
+
−
| | Let us find the '''right hand limit''' of this '''function''' as '''x''' tends to 2.
+
  
 
We will choose values of '''x''' greater than but close to 2.
 
We will choose values of '''x''' greater than but close to 2.
−
 
  
 
In '''column A''' from '''cells''' 6 to 10, type 2.01, 2.03, 2.05, 2.07 and 2.09.
 
In '''column A''' from '''cells''' 6 to 10, type 2.01, 2.03, 2.05, 2.07 and 2.09.
 
|-
 
|-
−
| | In '''cell B1''' (that is, '''column B, cell 1'''), type '''(3(A1)^2-A1-10)/((A1)^2-4)''' >> '''Enter'''.
+
||In '''cell B1''' (that is, '''column B, cell 1'''), type '''(3(A1)^2-A1-10)/((A1)^2-4)''' >> '''Enter'''.
−
| | In '''cell B1''' (that is, '''column B, cell 1'''), type the following ratio of values.
+
||In '''cell B1''' (that is, '''column B, cell 1'''), type the following ratio of values.
  
 
First, the numerator in parentheses
 
First, the numerator in parentheses
Line 299: Line 144:
 
'''A1''' in parentheses '''caret''' 2 minus 4  and press '''Enter'''.
 
'''A1''' in parentheses '''caret''' 2 minus 4  and press '''Enter'''.
 
|-
 
|-
−
| | Click on '''cell B1''' to highlight it.
+
||Click on '''cell B1''' to highlight it.
  
 
Place the '''cursor''' at the bottom right corner of the '''cell'''.
 
Place the '''cursor''' at the bottom right corner of the '''cell'''.
Line 307: Line 152:
  
 
Point to '''y''' values in '''column B''' and to the '''x''' values in '''column A'''.
 
Point to '''y''' values in '''column B''' and to the '''x''' values in '''column A'''.
−
| | Click on '''cell B1''' to highlight it.
+
||Click on '''cell B1''' to highlight it.
  
 
Place the '''cursor''' at the bottom right corner of the '''cell'''.
 
Place the '''cursor''' at the bottom right corner of the '''cell'''.
Line 316: Line 161:
 
This fills in '''y''' values corresponding to the '''x''' values in '''column A'''.
 
This fills in '''y''' values corresponding to the '''x''' values in '''column A'''.
 
|-
 
|-
−
| | Drag and increase column width.
+
||Drag and increase column width.
−
| | Drag and increase column width.
+
||Drag and increase column width.
 
|-
 
|-
−
| | Point to the '''question mark''' in '''cell B5''' corresponding to '''x=2'''.
+
||Point to the '''question mark''' in '''cell B5''' corresponding to '''x=2'''.
  
−
Point to the spreadsheet.
+
||Note that a question mark appears in '''cell B5''' corresponding to '''x equals 2'''.
  
−
| | Note that a question mark appears in '''cell B5''' corresponding to '''x equals 2'''.
+
This is because the '''function''' is undefined at this value.
 +
|-
 +
||Point to the spreadsheet.
 +
||Observe that as '''x''' tends to 2, '''y''' tends to 2.75.
  
  
−
This is because the '''function''' is undefined at this value.
+
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.
  
−
The reason for this is that the denominator of the '''function''' becomes 0.
+
|-
 +
||'''Slide Number 7'''
  
 +
'''Limits of discontinuous functions'''
  
−
Observe that as '''x''' tends to 2, '''y''' tends to 2.75.
+
'''lim h(x) = ?'''
  
 +
'''x→c'''
  
−
Hence, as '''x''' tends to 2, the limit of the '''function''' tends to 2.75.
+
'''lim h(x) = L4; lim h(x) = L3'''
−
|-
+
−
| | '''Slide Number 12'''
+
  
−
'''Limit of a rational polynomial function'''
+
'''x→c- x→c+'''
 +
Thus, '''lim h(x)''' Does Not Exist ('''DNE''')
  
−
'''lim_(x→2) (3x<sup>2</sup> – x -10)/(x<sup>2</sup> – 4) = 2.75'''
+
'''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''').
  
−
Thus, the '''limit''' of this '''rational polynomial function''' as '''x''' tends to 2 is 2.75.
 
 
|-
 
|-
−
| | '''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'''
−
| |
+
||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 369: Line 238:
 
We want to find the limits when '''x''' tends to 0 and 1.
 
We want to find the limits when '''x''' tends to 0 and 1.
 
|-
 
|-
−
| | Open a new '''GeoGebra''' window.
+
||Open a new '''GeoGebra''' window.
−
| | Let us open a new '''GeoGebra''' window.
+
||Let us open a new '''GeoGebra''' window.
 
|-
 
|-
−
| | Type '''a=Function[2x+3,-5,0]''' in the '''input bar''' >> '''Enter'''
+
||Type '''a=Function[2x+3,-5,0]''' in the '''input bar''' >> '''Enter'''
−
 
+
|| In the '''input bar''', type the following line.
−
 
+
−
 
+
−
| | In the '''input bar''', type the following line.
+
−
 
+
−
'''a''' equals '''Function''' with capital F and in square brackets '''2x plus 3''' comma minus 5 comma 0'''
+
  
  
Line 385: Line 249:
 
Press '''Enter'''.
 
Press '''Enter'''.
 
|-
 
|-
−
| | Point to the equation '''a(x)=2x+3 (-5 ≤ x ≤ 0)''' in '''Algebra''' view.
+
||Point to the equation '''a(x)=2x+3 (-5 ≤ x ≤ 0)''' in '''Algebra''' view.
−
 
+
  
 
Drag the boundary to see it properly.
 
Drag the boundary to see it properly.
−
 
  
 
Point to its graph in '''Graphics''' view.
 
Point to its graph in '''Graphics''' view.
−
| | The equation '''a of x equals 2x plus 3''' where '''x''' varies from minus 5 to 0 appears in '''Algebra''' view.
+
||The equation '''a of x equals 2x plus 3''' where '''x''' varies from minus 5 to 0 appears in '''Algebra''' view.
 +
 
  
 
Drag the boundary to see it properly.
 
Drag the boundary to see it properly.
Line 398: Line 261:
 
Its graph is seen in '''Graphics''' view.
 
Its graph is seen in '''Graphics''' view.
 
|-
 
|-
−
| | Under '''Move Graphics View''', click on '''Zoom Out''' and click in '''Graphics''' view.
+
||Under '''Move Graphics View''', click on '''Zoom Out''' and click in '''Graphics''' view.
−
 
+
|| Under '''Move Graphics View''', click on '''Zoom Out''' and click in '''Graphics''' view.
−
| | Under '''Move Graphics View''', click on '''Zoom Out''' and click in '''Graphics''' view.
+
  
 
|-
 
|-
−
| | Click on '''Move Graphics View''' and drag the background to see the graph properly.
+
||Click on '''Move Graphics View''' and drag the background to see the graph properly.
−
| | Click on '''Move Graphics View''' and drag the background to see the graph properly.
+
||Click on '''Move Graphics View''' and drag the background to see the graph properly.
 
|-
 
|-
−
| | Click on '''Move Graphics View''' tool, place '''cursor''' on '''x-axis'''.
+
||Click on '''Move Graphics View''' tool, place '''cursor''' on '''x-axis'''.
  
 
When an arrow appears along the '''axis''', drag the '''x-axis''' to zoom in or out.
 
When an arrow appears along the '''axis''', drag the '''x-axis''' to zoom in or out.
−
| | Click on '''Move Graphics View''' and place the '''cursor''' on the '''x-'axis'''.
+
||Click on '''Move Graphics View''' and place the '''cursor''' on the '''x-'axis'''.
  
 
When an arrow appears along the '''axis''', drag the '''x-axis''' to zoom in or out.
 
When an arrow appears along the '''axis''', drag the '''x-axis''' to zoom in or out.
 
|-
 
|-
−
| | Similarly, click on '''Move Graphics View''' tool and place '''cursor''' on '''y-axis'''.
+
||Click on '''Move Graphics View''' tool and place '''cursor''' on '''y-axis'''.
  
 
When an arrow appears along the '''axis''', drag the '''y-axis''' to zoom in or out.
 
When an arrow appears along the '''axis''', drag the '''y-axis''' to zoom in or out.
−
| | Similarly, click on '''Move Graphics View''' and place the '''cursor''' on the '''y-axis'''.
+
||Similarly, place the '''cursor''' on the '''y-axis'''.
 +
 
  
 
When an arrow appears along the '''axis''', drag the '''y-axis''' to zoom in or out.
 
When an arrow appears along the '''axis''', drag the '''y-axis''' to zoom in or out.
 
|-
 
|-
−
| | Click in and drag the background to see the graph properly.
+
||Click in and drag the background to see the graph properly.
−
| | Click in and drag the background to see the graph properly.
+
||Click in and drag the background to see the graph properly.
 
|-
 
|-
−
| | Type '''b=Function[3(x+1),0.01,5]''' in the '''input bar''' >> '''Enter'''
+
||Type '''b=Function[3(x+1),0.01,5]''' in the '''input bar''' >> '''Enter'''
  
−
| | In the '''input bar''', type the following command and press '''Enter'''.
+
||In the '''input bar''', type the following command.
  
−
Remember the space denotes multiplication.
 
  
−
'''b''' equals '''Function''' with capital F
+
Remember the space denotes multiplication.
−
 
+
−
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.
  
 
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'''. 
 +
|-
 +
||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.
 
|-
 
|-
−
| | 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'''.
−
| | 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.
 
|-
 
|-
−
| | Close the '''Preferences''' dialog box.
+
||Close the '''Preferences''' dialog box.
−
| | 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'''.
−
| | Under '''Move Graphics View''', click on '''Zoom In'''.
+
  
−
Now click on '''Move Graphics View''' and drag the background until you can see both graphs.
+
Click on '''Move Graphics View''' and drag the background
 +
||Under '''Move Graphics View''', click on '''Zoom In'''
 +
and click in '''Graphics''' view to magnify the graph.
 +
|-
 +
||click on '''Move Graphics View''' >>
 +
Drag the background to see both graphs.
 +
||Again click on '''Move Graphics View''' and drag the background until you can see both graphs.
 
|-
 
|-
−
| | Point to the break between the blue and red '''functions''' for '''f(x)=3(x+1).'''
+
||Point to the break between the blue and
−
| | Note that there is a break between the blue and red '''functions'''.
+
red functions.
 +
||Continue to '''Zoom In''' and drag the background
 +
until you see the gap between the functions.
  
 
This is because '''x''' is not 0 when '''f of x''' is '''3 times x plus 1'''.
 
This is because '''x''' is not 0 when '''f of x''' is '''3 times x plus 1'''.
 
|-
 
|-
−
| | Point to the blue '''function'''.
+
||Point to the red function.
−
+
||The red '''function''' has to be considered for '''x''' less
−
Point to intersection of '''f(x)''' and '''y-axis''' at '''(0,3)'''.
+
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.
 
  
−
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'''
+
||Point to the blue function.
 +
||The blue '''function''' has to be considered for '''x''' greater than 0.
  
−
..........................= '''3(x+1), x > 0'''
+
When '''x''' equals 1, the value of '''f of x''' is 6.
−
 
+
−
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'''
−
| | In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:
+
||In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:
  
 
Understand limits of '''functions'''
 
Understand limits of '''functions'''
  
 
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>
  
−
| | '''As an Assignment''':
+
'''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 '''rational polynomial function''' as '''x''' tends to 2.
Line 527: Line 388:
 
Find the limit of this '''trigonometric function''' as '''x''' tends to 0.
 
Find the limit of this '''trigonometric function''' as '''x''' tends to 0.
 
|-
 
|-
−
| | '''Slide Number 17'''
+
||'''Slide Number 11'''
  
 
'''About Spoken Tutorial project'''
 
'''About Spoken Tutorial project'''
−
| | The video at the following link summarizes the '''Spoken Tutorial''' project.
+
||The video at the following link summarizes the '''Spoken Tutorial''' project.
  
 
Please download and watch it.
 
Please download and watch it.
 
|-
 
|-
−
| | '''Slide Number 18'''
+
||'''Slide Number 12'''
  
 
'''Spoken Tutorial workshops'''
 
'''Spoken Tutorial workshops'''
−
| | The '''Spoken Tutorial Project''' team:
+
||The '''Spoken Tutorial Project''' team:
  
 
<nowiki>* conducts workshops using spoken tutorials and</nowiki>
 
<nowiki>* conducts workshops using spoken tutorials and</nowiki>
Line 545: Line 406:
 
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 558: Line 419:
  
 
Someone from our team will answer them
 
Someone from our team will answer them
−
| | Please post your timed queries on this forum.
+
||Please post your timed queries on this forum.
 
|-
 
|-
−
| | '''Slide Number 20'''
+
||'''Slide Number 14'''
  
 
'''Acknowledgement'''
 
'''Acknowledgement'''
−
| | '''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.
+
||'''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.
  
 
More information on this mission is available at this link.
 
More information on this mission is available at this link.
 
|-
 
|-
 
| |
 
| |
−
| | This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.
+
||This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.
  
 
Thank you for joining.
 
Thank you for joining.
 
|-
 
|-
 
|}
 
|}

Latest revision as of 22:42, 11 December 2018

Visual Cue Narration
Slide Number 1

Title Slide

Welcome to this tutorial on Limits and Continuity of Functions.
Slide Number 2

Learning Objectives

In this tutorial, we will learn how to use GeoGebra to:

Understand limits of functions

Look at continuity of functions

Slide Number 3

System Requirement

Here I am using:

Ubuntu Linux OS version 16.04

GeoGebra 5.0.481.0-d

Slide Number 4

Pre-requisites

www.spoken-tutorial.org

To follow this tutorial, you should be familiar with:

GeoGebra interface

Limits

Elementary calculus

For relevant tutorials, please visit our website.

Slide Number 5

Limits

Limits

Let us understand the concept of limits.

Imagine yourself sliding along the curve or line towards a given value of x.

The height at which you will be, is the corresponding y value of the function.

Any value of x can be approached from two sides.

The left side gives the left hand limit.

The right side gives the right hand limit.

Slide Number 6

Limit of a rational polynomial function

Let us find lim (3x2 – x -10)

x→2 (x2 – 4)

Limit of a rational polynomial function

Let us find the limit of this rational polynomial function as x tends to 2.

Show the GeoGebra window. I have already opened the GeoGebra interface.
To type the caret symbol, hold the Shift key down and press 6.

Type (3 x^2-x-10)/(x^2-4) in the input bar >> Enter

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

Note that spaces denote multiplication.


In the input bar, first type the numerator.

Now, type the denominator.

Press Enter.

Point to the equation in Algebra view and its graph in Graphics view. The equation appears in Algebra view and its graph in Graphics view.
Drag the boundary. Drag the boundary to see both properly.
Click on Move Graphics View tool.

Click in and drag Graphics view to see the graph.

Click on Move Graphics View.

Click in and drag Graphics view to see the graph.

Point to the graph in Graphics view. As x approaches 2, the function approaches some value close to 3.
Click on View tool >> select Spreadsheet. Click on View and select Spreadsheet.
Point to the spreadsheet on the right side of the Graphics view. This opens a spreadsheet on the right side of the Graphics view.
Click on Options tool and click on Rounding and choose 5 decimal places. Click on Options and click on Rounding and choose 5 decimal places.
Remember to press Enter to go to the next cell.

Type 1.91, 1.93, 1.96, 1.98 and 2 in column A from cells 1 to 5.

Let us find the left hand limit of this function as x tends to 2.

We will choose values of x less than but close to 2.

Remember to press Enter to go to the next cell.

In column A in cells 1 to 5, type 1.91, 1.93, 1.96, 1.98 and 2.

Type 2.01, 2.03, 2.05, 2.07 and 2.09 in column A from cells 6 to 10. Let us find the right hand limit of this function as x tends to 2.

We will choose values of x greater than but close to 2.

In column A from cells 6 to 10, type 2.01, 2.03, 2.05, 2.07 and 2.09.

In cell B1 (that is, column B, cell 1), type (3(A1)^2-A1-10)/((A1)^2-4) >> Enter. In cell B1 (that is, column B, cell 1), type the following ratio of values.

First, the numerator in parentheses

3 A1 in parentheses caret 2 minus A1 minus 10 followed by division slash

Now the denominator in parentheses

A1 in parentheses caret 2 minus 4 and press Enter.

Click on cell B1 to highlight it.

Place the cursor at the bottom right corner of the cell.


Drag the cursor to highlight cells until B10.

Point to y values in column B and to the x values in column A.

Click on cell B1 to highlight it.

Place the cursor at the bottom right corner of the cell.

Drag the cursor to highlight cells until B10.


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

Drag and increase column width. Drag and increase column width.
Point to the question mark in cell B5 corresponding to x=2. Note that a question mark appears in cell B5 corresponding to x equals 2.

This is because the function is undefined at this value.

Point to the spreadsheet. Observe that as x tends to 2, y tends to 2.75.


Hence, as x tends to 2, the limit of the function tends to 2.75.

Click in Graphics view and drag the background

to see this properly.

Click in Graphics view and drag the background

to see this properly.

Slide Number 7

Limits of discontinuous functions

lim h(x) = ?

x→c

lim h(x) = L4; lim h(x) = L3

x→c- x→c+ Thus, lim h(x) Does Not Exist (DNE)

x→c

In graph B, h of x is a piecewise or discontinuous function.

We want to find the limit of h of x as x approaches c.

So let us look at the left and right hand limits.

For the left hand limit, look at the lower limb where the limit is L4.

For the right hand limit, look at the upper limb where limit of h of x is L3.

But as x approaches c, the two limbs of h of x approach different values of y.

These are L3 and L4.

The left and right hand limits exist.

But the limit of h of x as x approaches c, itself does not exist (DNE).

Slide Number 8

Limit of a discontinuous function


Let us find lim f(x) = 2x+3, x ≤ 0

x→0 3(x+1), x > 0

and lim f(x) = 2x+3, x ≤ 0

x→1 3(x+1), x > 0

Limit of a discontinuous function.

Let us find limits of a piecewise or discontinuous function f of x.


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

But f of x is described by 3 times x plus 1 when x is greater than 0.

We want to find the limits when x tends to 0 and 1.

Open a new GeoGebra window. Let us open a new GeoGebra window.
Type a=Function[2x+3,-5,0] in the input bar >> Enter In the input bar, type the following line.


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

Press Enter.

Point to the equation a(x)=2x+3 (-5 ≤ x ≤ 0) in Algebra view.

Drag the boundary to see it properly.

Point to its graph in Graphics view.

The equation a of x equals 2x plus 3 where x varies from minus 5 to 0 appears in Algebra view.


Drag the boundary to see it properly.

Its graph is seen in Graphics view.

Under Move Graphics View, click on Zoom Out and click in Graphics view. Under Move Graphics View, click on Zoom Out and click in Graphics view.
Click on Move Graphics View and drag the background to see the graph properly. Click on Move Graphics View and drag the background to see the graph properly.
Click on Move Graphics View tool, place cursor on x-axis.

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

Click on Move Graphics View and place the cursor on the x-'axis.

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

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

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

Similarly, place the cursor on the y-axis.


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

Click in and drag the background to see the graph properly. Click in and drag the background to see the graph properly.
Type b=Function[3(x+1),0.01,5] in the input bar >> Enter In the input bar, type the following command.


Remember the space denotes multiplication.

This chooses the domain of x from 5 (for practical purposes) to 0.01.

For this piece of the function, x is greater than 0 but not equal to 0.

Press Enter.

Drag the boundary to see the equation properly. Drag the boundary to see the equation properly.
Point to the equation b(x)=3(x+1) (0.01 ≤ x ≤ 5) in Algebra view.

Point to its graph in Graphics view.

The equation b of x equals 3 times x plus 1 where x varies from 0.01 to 5 appears in Algebra view.


Its graph appears in Graphics view.

Double click on the equation a(x)=2x+3 in Algebra view. In Algebra view, double click on the equation b of x equals 3 times x plus 1.
Click on Object Properties. Click on Object Properties.
Click on Color tab >> select blue. Click on the Color tab and select blue.
Close the Preferences dialog box. Close the Preferences dialog box.
Click in and drag the background. Click in and drag the background to see both functions in Graphics view.
Under Move Graphics View, click on Zoom In.

Click on Move Graphics View and drag the background

Under Move Graphics View, click on Zoom In

and click in Graphics view to magnify the graph.

click on Move Graphics View >>

Drag the background to see both graphs.

Again click on Move Graphics View and drag the background until you can see both graphs.
Point to the break between the blue and

red functions.

Continue to Zoom In and drag the background

until you see the gap between the functions.

This is because x is not 0 when f of x is 3 times x plus 1.

Point to the red function. The red function has to be considered for x less

than and equal to 0.

When x tends to 0, f of x is 3 as the function intersects the y-axis at 0 comma 3.


Point to the blue function. The blue function has to be considered for x greater than 0.

When x equals 1, the value of f of x is 6.

Let us summarize.
Slide Number 9

Summary

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

Understand limits of functions

Look at continuity of functions


Slide Number 10

Assignment


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

Evaluate lim sin4x

x→0 sin 2x

As an Assignment:

Find the limit of this rational polynomial function as x tends to 2.

Find the limit of this trigonometric function as x tends to 0.

Slide Number 11

About Spoken Tutorial project

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

Please download and watch it.

Slide Number 12

Spoken Tutorial workshops

The Spoken Tutorial Project team:

* conducts workshops using spoken tutorials and

* gives certificates on passing online tests.

For more details, please write to us.

Slide Number 13

Forum for specific questions:

Do you have questions in THIS Spoken Tutorial?

Please visit this site

Choose the minute and second where you have the question

Explain your question briefly

Someone from our team will answer them

Please post your timed queries on this forum.
Slide Number 14

Acknowledgement

Spoken Tutorial Project is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.

This is Vidhya Iyer from IIT Bombay, signing off.

Thank you for joining.

Contributors and Content Editors

Madhurig, Vidhya

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