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:

How close should x be to 3 so that is within 0.1 of 2?

The x-Tolerance

The procedure to find the solution to the question above can be expressed using two compound inequalities.

Find a positive number so that whenever

Solving the Inequality

When the y-tolerance is 0.1 the x-tolerance can be found by solving the
 A compound (or extended) inequality is an inequality that compares more than two quantities and contains more than one inequality symbol.
compound inequality, , as shown below.

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 y-Tolerance

The second x-tolerance you found in Lesson 6.1 came from answering the question

How close should x be to 3 to ensure that is within 0.01 of 2?

6.2.3 Compare the answer in 6.2.2 with the inequality 3 - < x < 3 + . Find a value of , the x-tolerance. Click here for the answer.

Finding a Generalized Solution

Suppose that you are asked to find values of that correspond to smaller and smaller y-tolerances around y = 2. Rather than going through the same process repeatedly for various y-tolerances, you could solve the problem once with a generalized y-tolerance.

Let the general y-tolerance 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 x-Tolerance

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:

and

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

The limit of the function as x approaches 3 is equal to 2.

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

whenever

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

whenever

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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