Module 4 Using TI-89 or Voyage 200 to Explore the Concept of Limit

Finding
Compare 2.87 < x < 3.13667 with the inequality 3 – < x < 3 + . It must follow that 3 – 2.87
and 3 + 3.13667 . However, solving these two equations for yields two different values:
0.13 and 0.13667. As in lesson 4.1, the smaller of the two values is the correct answer:
0.13.
Smaller Tolerance
The second tolerance you found in lesson 4.1 came from answering the question
How close should x be to 3 so that is within 0.01 of 2
4.2.1 Rephrase this question with compound inequalities.
4.2.2 Use the solve( command to solve the first compound inequality.
4.2.3 Compare your answer to 4.2.2 with the inequality 3 – < x < 3 + and find a value of , the
x-tolerance.
Finding a Generalized Solution
Suppose that you are asked to find values of that correspond to smaller and smaller tolerances
for y around y = 3. Rather than going through the same process over and over for various y-
tolerances, you could solve the problem once with a generalized y-tolerance. Call this general y-
tolerance , the Greek letter "epsilon." The Greek letters delta, , and epsilon, , will be used to
represent small positive numbers.
The tolerance question then becomes
How close should x be to 3 so that is within of 2
Rephrasing the Question
The question can be rephrased using inequalities as
Find a positive number so that 2 – < < 2 + whenever 3 – < x < 3 +
Solving the Left Inequality: 2 – <
• Solve the left side of the inequality 2 – < < 2 + with the command
solve(2-
= (3x–5),x)
The Greek letter can be entered in the TI-89 by pressing .