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Using the graph of the function shown in [link] , estimate the following limits.
a. 0; b. 2; c. does not exist; d. e. 0; f. does not exist; g. 4; h. 4; i. 4
Creating a table is a way to determine limits using numeric information. We create a table of values in which the input values of approach from both sides. Then we determine if the output values get closer and closer to some real value, the limit
Let’s consider an example using the following function:
To create the table, we evaluate the function at values close to We use some input values less than 5 and some values greater than 5 as in [link] . The table values show that when but nearing 5, the corresponding output gets close to 75. When but nearing 5, the corresponding output also gets close to 75.
Because
then
Remember that does not exist.
Given a function use a table to find the limit as approaches and the value of if it exists.
Numerically estimate the limit of the following expression by setting up a table of values on both sides of the limit.
We can estimate the value of a limit, if it exists, by evaluating the function at values near We cannot find a function value for directly because the result would have a denominator equal to 0, and thus would be undefined.
We create [link] by choosing several input values close to with half of them less than and half of them greater than Note that we need to be sure we are using radian mode. We evaluate the function at each input value to complete the table.
The table values indicate that when but approaching 0, the corresponding output nears
When but approaching 0, the corresponding output also nears
Because
then
Is it possible to check our answer using a graphing utility?
Yes. We previously used a table to find a limit of 75 for the function as approaches 5. To check, we graph the function on a viewing window as shown in [link] . A graphical check shows both branches of the graph of the function get close to the output 75 as nears 5. Furthermore, we can use the ‘trace’ feature of a graphing calculator. By appraoching we may numerically observe the corresponding outputs getting close to
Numerically estimate the limit of the following function by making a table:
Is one method for determining a limit better than the other?
No. Both methods have advantages. Graphing allows for quick inspection. Tables can be used when graphical utilities aren’t available, and they can be calculated to a higher precision than could be seen with an unaided eye inspecting a graph.
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