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Chapter 4 | Applications of Derivatives
Figure 4.48 For a function with a limit at infinity, for all x > N , | f ( x )− L | < ε .
Earlier in this section, we used graphical evidence in Figure 4.40 and numerical evidence in Table 4.2 to conclude that lim x →∞ ⎛ ⎝ 2+ 1 x ⎞ ⎠ =2. Here we use the formal definition of limit at infinity to prove this result rigorously.
Example 4.22 A Finite Limit at Infinity Example
Use the formal definition of limit at infinity to prove that lim x →∞ ⎛ ⎝ 2+ 1 x ⎞ ⎠ =2.
Solution Let ε >0. Let N = 1 ε . Therefore, for all x > N , we have | 2+ 1 x −2 | = | 1
x | = 1 x < 1 N =
ε .
Use the formal definition of limit at infinity to prove that lim x →∞ ⎛ ⎝ 3− 1 x 2 ⎞ ⎠ =3.
4.21
We now turn our attention to a more precise definition for an infinite limit at infinity.
Definition (Formal) We say a function f has an infinite limit at infinity and write lim x →∞ f ( x ) =∞ if for all M >0, there exists an N >0 such that f ( x ) > M for all x > N (see Figure 4.49 ). We say a function has a negative infinite limit at infinity and write lim x →∞ f ( x ) =−∞ if for all M <0, there exists an N >0 such that f ( x ) < M
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