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:
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The x-Tolerance
The procedure to find the solution to the question above can be expressed using two compound inequalities.
![]() ![]() ![]() Solving the Inequality
When the y-tolerance is 0.1 the x-tolerance 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
Solving the two equations for
A Smaller y-Tolerance The second x-tolerance 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 -
Finding a Generalized Solution
Suppose that you are asked to find values of
Let the general y-tolerance be represented by
The tolerance question then becomes
How close should x be to 3 so that
Finding the Generalized x-Tolerance The procedure used to find the solution to the generalized tolerance question is
Given a positive number
Solving the Generalized Inequality
The compound inequality
![]()
The solution to the generalized compound inequality is
Comparing the last inequality with
![]() ![]()
The solutions to these equations are
The smaller of these two values is
Given any small positive value for
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
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That is, the value of
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The limit is read "the limit of
In the previous example we found that
![]() ![]()
Because for any positive
![]() 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,
![]() ![]()
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
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