Limits via Tables
Document Actions
Let's take another look at some of the functions we have been exploring, but using a table of values in addition to the graph of the function. Each example includes a table of values of the function which approach c from the left and right.
Try the following:
- The first graph show the line used in a previous example. Is the limit L = 0.5 when c = 1 ? In other words, does f (x) approach 0.5 as x approaches 1? Looking at the
table of values, as x approaches 1 from either direction, the
output value of the function approaches 0.5. Note that the table does not
present a value for f (c) as this is not needed to find the
limit.
- Select the second example. This is just like the first case, except
that one point has moved. Is the limit still L = 0.5 when c = 1 ? In other words, does f (x) approach 0.5 as x approaches 1? Looking at the table of values, as x approaches
1 from either direction, the output value of the function approaches 0.5.
The fact that f (c) does not equal 0.5 (it equals 1) has no
effect on the limit.
- Select the third example. This is like the previous two cases, but
there is now a point missing. Is the limit still L = 0.5 when c = 1 ? In other words, does f (x) approach 0.5 as x approaches 1? Looking at the table of values, as x approaches 1 from either direction, the output value of the function
approaches 0.5. The fact that f (c) is undefined has no
effect on the limit.
- Select the fourth example. This is a more complex function, but this
example is similar to the previous one with a missing point. What is the
limit when c = 0 ? In other words, what value does f (x) approach as x approaches 0? As you can see from the
table of values, the output value approaches 1 from both directions.
- Select the fifth example, a jump discontinuity. What is the limit when c = 1? In other words, what value does f (x)
approach as x approaches 1? Note that in this case, the table
shows different values for the left-hand and right-hand limits. Hence
there is no general limit at c = 1.
- Select the sixth example, a function with a vertical asymptote. What is
the limit when c = 1? In other words, what value does f (x) approach as x approaches 1? The table shows that
the output value gets bigger and bigger as you approach 1 from either
direction, hence there is no limit.
- Select the seventh example, the wiggly sin(1/x). The table shows that the value jumps around as you approach 0, hence the limit does not exist there.
For more information on rights and downloading, refer to http://www.calculusapplets.com/download.html.
© Copyright 2001 David J. Eck
© Copyright 2007 Thomas S. Downey
Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the “Software”), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
You must cause any files that you modify to carry prominent notices stating that you changed the files and the date of any change, and modified files must be put into a Java package different from edu.hws.
THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.
