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Sullivan AP˙Sullivan˙Chapter01 October 8, 2016 17:4
Sullivan
148 Chapter 1 • Limits and Continuity
( ) c
So, given any ε>0, we choose δ = min
2 , c2
2 · ε . Then whenever 0 < |x−c| 0. ■ NOW WORK Problem 25.
The ε-δ definition of a limit can be used to show that a limit does not exist, or that a
limit is not equal to a specific number. Example 5 illustrates how the ε-δ definition of a
limit is used to show that a limit is not equal to a specific number.
NOTE In a proof by contradiction, we
assume that the conclusion is not true
and then show this leads to a
contradiction.
EXAMPLE 5
Showing a Limit Is Not Equal to a Specific Number
Use the ε-δ definition of a limit to prove the statement lim
x→3
(4x − 5) = 10.
Solution We use a proof by contradiction. Assume lim
x→3
(4x −5) = 10 and choose ε = 1.
(Any smaller positive number ε will also work.) Then there is a number δ>0, so that
whenever 0 < |x − 3|