|00:01||Welcome to this tutorial on Limits and Continuity of Functions.|
|00:07||In this tutorial, we will learn how to use GeoGebra to:
Understand limits of functions
|00:15||Look at continuity of functions|
|00:18||Here I am using:
Ubuntu Linux OS version 16.04
GeoGebra 5.0.481.0 hyphen d
|00:31||To follow this tutorial, you should be familiar with:|
|00:36||GeoGebra interface, Limits, Elementary calculus|
|00:42||For relevant tutorials, please visit our website.|
|00:48||Let us understand the concept of limits.|
|00:52||Imagine yourself sliding along the curve or line towards a given value of x.|
|01:00||The height at which you will be, is the corresponding y value of the function.|
|01:07||Any value of x can be approached from two sides.|
|01:12||The left side gives the left hand limit.
The right side gives the right hand limit.
|01:19||Limit of a rational polynomial function|
|01:23||Let us find the limit of this rational polynomial function as x tends to 2.|
|01:31||I have already opened the GeoGebra interface.|
|01:36||To type the caret symbol, hold the Shift key down and press 6.|
|01:42||Note that spaces denote multiplication.|
|01:46||In the input bar, first type the numerator.|
|01:50||Now, type the denominator.
|01:56||The equation appears in Algebra view and its graph in Graphics view.|
|02:03||Drag the boundary to see both properly.|
|02:08||Click on Move Graphics View.|
|02:12||Click in and drag Graphics view to see the graph.|
|02:21||As x approaches 2, the function approaches some value close to 3.|
|02:29||Click on View and select Spreadsheet.|
|02:34||This opens a spreadsheet on the right side of the Graphics view.|
|02:40||Click on Options and click on Rounding and choose 5 decimal places.|
|02:49||Let us find the left hand limit of this function as x tends to 2.|
|02:55||We will choose values of x less than but close to 2.|
|03:00||Remember to press Enter to go to the next cell.|
|03:04||In column A in cells 1 to 5, type 1.91, 1.93, 1.96, 1.98 and 2.|
|03:23||Let us find the right hand limit of this function as x tends to 2.|
|03:29||We will choose values of x greater than but close to 2.|
|03:35||In column A from cells 6 to 10, type 2.01, 2.03, 2.05, 2.07 and 2.09.|
|03:54||In cell B1 (that is, column B, cell 1), type the following ratio of values.|
|04:02||First, the numerator in parentheses
3 A1 in parentheses caret 2 minus A1 minus 10 followed by division slash
|04:18||Now the denominator in parentheses
A1 in parentheses caret 2 minus 4 and press Enter.
|04:28||Click on cell B1 to highlight it.|
|04:33||Place the cursor at the bottom right corner of the cell.|
|04:38||Drag the cursor to highlight cells until B10.|
|04:43||This fills in y values corresponding to the x values in column A.|
|04:49||Drag and increase column width.|
|04:53||Note that a question mark appears in cell B5 corresponding to x equals 2.|
|05:01||This is because the function is undefined at this value.|
|05:06||Observe that as x tends to 2, y tends to 2.75.|
|05:14||Hence, as x tends to 2, the limit of the function tends to 2.75.|
|05:22||Click in Graphics view and drag the background to see this properly.|
|05:31||Limits of Discontinuous Functions .|
|05:34||In graph B, h of x is a piecewise or discontinuous function.|
|05:43||We want to find the limit of h of x as x approaches c.|
|05:49||So let us look at the left and right hand limits.|
|05:43||For the left hand limit, look at the lower limb where the limit is L4.|
|06:00||For the right hand limit, look at the upper limb where limit of h of x is L3.|
|06:07||But as x approaches c, the two limbs of h of x approach different values of y.|
|06:16||These are L3 and L4.|
|06:20||The left and right hand limits exist.|
|06:24||But the limit of h of x as x approaches c, itself does not exist (DNE).|
|06:33||Limit of a discontinuous function.|
|06:36||Let us find limits of a piecewise or discontinuous function f of x.|
|06:43||f of x is described by 2x plus 3 when x is 0 or less than 0.|
|06:50||But f of x is described by 3 times x plus 1 when x is greater than 0.|
|06:59||We want to find the limits when x tends to 0 and 1.|
|07:07||Let us open a new GeoGebra window.|
|07:11||In the input bar, type the following line.|
|07:15||This chooses the domain of x from minus 5 (for practical purposes) to 0.
|07:24||The equation a of x equals 2x plus 3 where x varies from minus 5 to 0 appears in Algebra view.|
|07:35||Drag the boundary to see it properly.|
|07:39||Its graph is seen in Graphics view.|
|07:43||Under Move Graphics View, click on Zoom Out and click in Graphics view.|
|07:51||Click on Move Graphics View and drag the background to see the graph properly.|
|07:59||Click on Move Graphics View and place the cursor on the x-'axis.|
|08:07||When an arrow appears along the axis, drag the x-axis to zoom in or out.|
|08:15||Similarly, place the cursor on the y-axis.|
|08:20||When an arrow appears along the axis, drag the y-axis to zoom in or out.|
|08:28||Click in and drag the background to see the graph properly.|
|08:33||In the input bar, type the following command.|
|08:37||Remember the space denotes multiplication.|
|08:41||This chooses the domain of x from 5 (for practical purposes) to 0.01.|
|08:49||For this piece of the function, x is greater than 0 but not equal to 0.
|08:57||Drag the boundary to see the equation properly.|
|09:01||The equation b of x equals 3 times x plus 1 where x varies from 0.01 to 5 appears in Algebra view.|
|09:12||Its graph appears in Graphics view.|
|09:16||In Algebra view, double click on the equation b of x equals 3 times x plus 1.|
|09:23||Click on Object Properties.|
|09:26||Click on the Color tab and select blue.|
|09:31||Close the Preferences dialog box.|
|09:34||Click in and drag the background to see both functions in Graphics view.|
|09:41||Under Move Graphics View, click on Zoom In and click in Graphics view to magnify the graph.|
|09:51||Again click on Move Graphics View and drag the background until you can see both graphs.|
|10:00||Continue to Zoom In and drag the background until you see the gap between the functions.|
|10:10||This is because x is not 0 when f of x is 3 times x plus 1.|
|10:18||The red function has to be considered for x less than and equal to 0.|
|10:25||When x tends to 0, f of x is 3 as the function intersects the y-axis at 0 comma 3.|
|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.
|10:50||Let us summarize.|
|10:52||In this tutorial, we have learnt how to use GeoGebra to:
Understand limits of functions, Look at continuity of functions
|11:03||As an Assignment:
Find the limit of this rational polynomial function as x tends to 2.
|11:12||Find the limit of this trigonometric function as x tends to 0.|
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