View source for Applications-of-GeoGebra/C3/Limits-and-Continuity-of-Functions/English-timed


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||'''Time'''
||'''Narration'''

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||00:01
||Welcome to this tutorial on '''Limits and Continuity of Functions'''.

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||00:07
||In this '''tutorial''', we will learn how to use '''GeoGebra''' to:

Understand '''limits''' of '''functions'''

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||00:15
|Look at continuity of '''functions'''

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||Here I am using:

'''Ubuntu Linux''' OS version 16.04

'''GeoGebra''' 5.0.481.0 hyphen d

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||00:31
||To follow this tutorial, you should be familiar with:

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||00:36
||'''GeoGebra''' interface, '''Limits''', '''Elementary calculus'''

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||00:42
|For relevant '''tutorials''', please visit our website.
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||00:46
||'''Limits'''

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||00:48
||Let us understand the concept of '''limits'''.

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||00:52
||Imagine yourself sliding along the curve or line towards a given value of '''x'''.

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

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||Any value of '''x''' can be approached from two sides.

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||The left side gives the '''left hand limit'''.

The right side gives the '''right hand limit'''.
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||'''Limit of a rational polynomial function'''

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||01:23
||Let us find the '''limit''' of this '''rational polynomial function''' as '''x''' tends to 2.
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||01:31
||I have already opened the '''GeoGebra''' interface.

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

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||Note that spaces denote multiplication.

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||In the '''input bar''', first type the '''numerator'''.

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

Press '''Enter'''.
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||The equation appears in '''Algebra''' view and its graph in '''Graphics''' view.
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||Drag the boundary to see both properly.
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||Click on '''Move Graphics View'''.

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||Click in and drag '''Graphics''' view to see the graph.
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||As '''x''' approaches 2, the '''function''' approaches some value close to 3.
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||Click on '''View''' and select '''Spreadsheet'''.
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||This opens a spreadsheet on the right side of the '''Graphics''' view.
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||Click on '''Options''' and click on '''Rounding''' and choose '''5 decimal places'''.
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||Let us find the '''left hand limit''' of this '''function''' as '''x''' tends to 2.

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

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||Remember to press '''Enter''' to go to the next '''cell'''.

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||03:04
||In '''column A''' in '''cells''' 1 to 5, type 1.91, 1.93, 1.96, 1.98 and 2.
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||03:23
||Let us find the '''right hand limit''' of this '''function''' as '''x''' tends to 2.

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||03:29
||We will choose values of '''x''' greater than but close to 2.

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||In '''column A''' from '''cells''' 6 to 10, type 2.01, 2.03, 2.05, 2.07 and 2.09.
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||In '''cell B1''' (that is, '''column B, cell 1'''), type the following ratio of values.

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||First, the numerator in parentheses

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

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||Now the denominator in parentheses

'''A1''' in parentheses '''caret''' 2 minus 4  and press '''Enter'''.
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||04:28
||Click on '''cell B1''' to highlight it.

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

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||Drag the '''cursor''' to highlight cells until '''B10'''.

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||This fills in '''y''' values corresponding to the '''x''' values in '''column A'''.
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||Drag and increase column width.
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||Note that a question mark appears in '''cell B5''' corresponding to '''x equals 2'''.

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||This is because the '''function''' is undefined at this value.
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||Observe that as '''x''' tends to 2, '''y''' tends to 2.75.

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

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||Click in Graphics view and drag the background to see this properly.

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|| Limits of Discontinuous Functions . 

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||05:34
||In graph '''B''', '''h of x''' is a '''piecewise''' or '''discontinuous function'''.

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

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

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

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

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

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||These are '''L3''' and '''L4'''.

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||The '''left''' and '''right hand limits''' exist.

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

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||Limit of a discontinuous function.

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

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||06:43
||'''f of x''' is described by '''2x plus 3''' when '''x''' is 0 or less than 0.

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

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||We want to find the limits when '''x''' tends to 0 and 1.
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|07:07
||Let us open a new '''GeoGebra''' window.
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|| In the '''input bar''', type the following line.

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

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

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||Drag the boundary to see it properly.

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||Its graph is seen in '''Graphics''' view.
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|| Under '''Move Graphics View''', click on '''Zoom Out''' and click in '''Graphics''' view.

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||Click on '''Move Graphics View''' and drag the background to see the graph properly.
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||Click on '''Move Graphics View''' and place the '''cursor''' on the '''x-'axis'''.

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||When an arrow appears along the '''axis''', drag the '''x-axis''' to zoom in or out.
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||Similarly, place the '''cursor''' on the '''y-axis'''.

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||When an arrow appears along the '''axis''', drag the '''y-axis''' to zoom in or out.
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||Click in and drag the background to see the graph properly.

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||In the '''input bar''', type the following command.

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||Remember the space denotes multiplication.

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

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

Press '''Enter'''.  
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||Drag the boundary to see the equation properly.
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||The equation '''b of x equals 3 times x plus 1''' where '''x''' varies from 0.01 to 5 appears in '''Algebra''' view.

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||Its graph appears in '''Graphics''' view.
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||In '''Algebra''' view, double click on the equation '''b of x''' equals 3 times '''x''' plus 1.
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||Click on '''Object Properties'''.
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||Click on the '''Color''' tab and select blue.
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||Close the '''Preferences''' dialog box.
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||Click in and drag the background to see both '''functions''' in '''Graphics''' view.
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||Under '''Move Graphics View''', click on '''Zoom In''' and click in '''Graphics''' view to magnify the graph.
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||09:51
||Again click on '''Move Graphics View''' and drag the background until you can see both graphs.
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||Continue to '''Zoom In''' and drag the background until you see the gap between the functions.

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||This is because '''x''' is not 0 when '''f of x''' is '''3 times x plus 1'''.
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||The red '''function''' has to be considered for '''x''' less than and equal to 0.

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||10:25
||When '''x''' tends to 0, '''f of x''' is 3 as the '''function''' intersects the '''y-axis''' at 0 comma 3.

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||10:35
||The blue '''function''' has to be considered for '''x''' greater than 0.

When '''x''' equals 1, the value of '''f of x''' is 6.
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||10:50
||Let us summarize.
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||10:52
||In this '''tutorial''', we have learnt how to use '''GeoGebra''' to:

Understand limits of '''functions''', Look at continuity of '''functions'''

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||11:03
||'''As an Assignment''':

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

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||11:12
||Find the limit of this '''trigonometric function''' as '''x''' tends to 0.
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||11:19
||The video at the following link summarizes the '''Spoken Tutorial''' project.

Please download and watch it.
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||11:27
||The '''Spoken Tutorial Project''' team: conducts workshops using spoken tutorials and

gives certificates on passing online tests.

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||11:36
||For more details, please write to us.
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||11:39
||Please post your timed queries on this forum.
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||11:43
||'''Spoken Tutorial Project''' is funded by NMEICT, MHRD, Government of India.

More information on this mission is available at this link.
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||11:56
||This is '''Vidhya Iyer''' from '''IIT Bombay,''' signing off.

Thank you for joining.
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