Module 4 - Limit as x Approaches a | ||||||||||||||||||||
Introduction | Lesson 1 | Lesson 2 | Lesson 3 | Self-Test | ||||||||||||||||||||
Lesson 4.2: Definition of Limit | ||||||||||||||||||||
In the previous lesson you found tolerances graphically. In this lesson you will use the TI-89 computer algebra system to find these tolerances symbolically. This will prepare you to generalize the tolerances and develop the definition of limit. Tolerances The first tolerance you found in lesson 4.1 was in response to the question
![]() Restate the Question Another way to phrase this question is
![]() ![]() ![]() ![]() Given y-Tolerance, Find x-Tolerance
The x-tolerance when the y-tolerance is 0.1 can be found by solving the
![]() The solution is approximately 2.87 < x < 3.13667. Notice that the solutions are the same as those found in Lesson 4.1 when you used the intersection feature.
Finding
Compare 2.87 < x < 3.13667 with the inequality 3
Smaller Tolerance The second tolerance you found in lesson 4.1 came from answering the question
![]() 4.2.1 Rephrase this question with compound inequalities. Click here for the answer. 4.2.2 Use the solve( command to solve the first compound inequality. Click here for the answer.
4.2.3 Compare your answer to 4.2.2 with the inequality 3
Finding a Generalized Solution
Suppose that you are asked to find values of
The tolerance question then becomes
![]() ![]() Rephrasing the Question The question can be rephrased using inequalities as
![]() ![]() ![]() ![]() ![]() ![]()
Solving the Left Inequality: 2
![]() ![]()
The Greek letter
![]() The solution to the equation has two parts:
The second part of the solution,
Solving the Right Inequality:
You can edit the command shown in the Edit Line to solve the right inequality by changing the "" sign to "+."
![]()
The solution is
Because
Combining the solutions to find
Combining the two solutions, in order to have 2
![]()
Comparing this inequality with 3
![]() ![]() Solving these equations
for
The smaller of these two values is
Limits When the values of the output can be made as close as we like to 2 by taking input values sufficiently close to 3, we say
![]()
That is, the value of f(x) =
![]()
which is read "the limit of
In the previous example we found that
![]() ![]() ![]() ![]() ![]()
Because for any positive
![]() Definition of Limit
Formally,
![]() ![]() ![]() ![]()
Conceptually, f(x)
4.2.4 Write the inequality with conditions that is associated with the limit. Interpret
Click here for the answer. |
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