Module 6  Limit as x Approaches a  
Introduction  Lesson 1  Lesson 2  Lesson 3  Self Test  
Lesson 6.2: Definition of Limit  
In the previous lesson you found tolerances graphically. In this lesson you will use algebra to find the tolerances symbolically. This will prepare you to generalize the tolerances and develop the definition of limit. Tolerances The first tolerance you found in lesson 6.1 was in response to the question:
The xTolerance The procedure to find the solution to the question above can be expressed using two compound inequalities.
Solving the Inequality
When the ytolerance is 0.1 the xtolerance can be found by solving the
The sense of the compound inequality above was retained when each expression was squared because each expression represented a positive number. The solution of the compound inequality is approximately 2.87 < x < 3.136666667. Notice that the solutions are the same as those found in Lesson 6.1 when you used the Intersect feature on the Graph screen. Finding Compare the inequalities below. It follows that and . Solving the two equations for yields two different values: and . As in Lesson 6.1, the smaller of the two values is the correct answer: . A Smaller yTolerance The second xtolerance you found in Lesson 6.1 came from answering the question
6.2.1 Restate the question above using compound inequalities. Click here for the answer. 6.2.2 Solve the first compound inequality in the answer to 6.2.1. Click here for the answer. 6.2.3 Compare the answer in 6.2.2 with the inequality 3  < x < 3 + . Find a value of , the xtolerance. Click here for the answer. Finding a Generalized Solution Suppose that you are asked to find values of that correspond to smaller and smaller ytolerances around y = 2. Rather than going through the same process repeatedly for various ytolerances, you could solve the problem once with a generalized ytolerance. Let the general ytolerance be represented by , the Greek letter "epsilon." The Greek letters delta, , and epsilon, , are used to represent small positive numbers. The tolerance question then becomes How close should x be to 3 so that is within of 2? Finding the Generalized xTolerance The procedure used to find the solution to the generalized tolerance question is Given a positive number , find a positive number , so that whenever Solving the Generalized Inequality The compound inequality is solved below. The method of solution assumes that is less than 2. If is greater than 2, we can choose = 1. The solution to the generalized compound inequality is Comparing the last inequality with yields two equations: The solutions to these equations are and If is a small positive number will be positive. The smaller of these two values is , which is the general solution. Given any small positive value for the last equation can be used to find the corresponding value for . For example, when = 0.01, this equation gives = 0.0133, the same value found earlier. 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 gets closer to 2 as x gets closer to 3. The notation used to indicate this is The limit is read "the limit of as x approaches 3 is 2." In the previous example we found that guarantees that Because for any positive a corresponding positive can be found that meets the conditions above, This leads to the formal definition of a limit. Definition of Limit The value of the function when x = a is not relevant when finding a limit. Only the values of the function are considered as x approaches a. Formally, if for any , there exists a such that Conceptually, f(x) approaches L as x approaches a, which can also be written The value of the function when x = a is not relevant when finding a limit. Only the values of the function as x approaches a are considered. 6.2.4 Write a statement with compound inequalities that represents the conditions associated with . Click here for the answer. 

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