Chapter 2 | Limits
195
Definition
Translation
1. For every ε >0,
1. For every positive distance ε from L ,
2. there exists a δ >0,
2. There is a positive distance δ from a ,
3. such that
3. such that
4. if 0< | x − a | < δ , then | f ( x )− L | < ε .
4. if x is closer than δ to a and x ≠ a , then f ( x ) is closer than ε to L .
Table 2.9 Translation of the Epsilon-Delta Definition of the Limit
We can get a better handle on this definition by looking at the definition geometrically. Figure 2.39 shows possible values of δ for various choices of ε >0 for a given function f ( x ), a number a , and a limit L at a . Notice that as we choose smaller values of ε (the distance between the function and the limit), we can always find a δ small enough so that if we have chosen an x value within δ of a , then the value of f ( x ) is within ε of the limit L .
Figure 2.39 These graphs show possible values of δ , given successively smaller choices of ε .
Visit the following applet to experiment with finding values of δ for selected values of ε : • The epsilon-delta definition of limit (http://www.openstax.org/l/20_epsilondelt)
Example 2.39 shows how you can use this definition to prove a statement about the limit of a specific function at a specified value. Example 2.39 Proving a Statement about the Limit of a Specific Function
(2 x +1) =3.
Prove that lim x →1
Solution
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