Applications-of-GeoGebra/C3/Limits-and-Continuity-of-Functions/English-timed
From Script | Spoken-Tutorial
Time | Narration |
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. |
00:46 | Limits |
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.
Press Enter. |
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.
Press Enter. |
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.
Press Enter. |
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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11:56 | This is Vidhya Iyer from IIT Bombay, signing off.
Thank you for joining. |