Introduction to Mathematical Analysis I - 3rd Edition

72 3.2 Limit Theorems Note that f (x) = 1 x2 >Mis equivalent to 1 M >x 2 since M>0. It follows that q1 M >|x|. Now, if we choose δ such that 0 <δ <q1 M. Then, if 0 <|x| <δ we get 0 <x 2 <δ2 and δ2 < 1 M. Therefore 1 x2 > 1 δ2 > 1 1 M =M, as desired. Definition 3.2.6 (limits at infinity) Let f : D→R. (i) When Dis not bounded above, we say that f has limit ℓ as x →∞if for every ε >0, there exists M∈Rsuch | f (x)−ℓ| <ε for all x >M,x ∈D and write limx→∞ f (x) =ℓ, (ii) WhenDis not bounded below, we say that f has limit ℓ as x →−∞if for everyε >0, there exists L∈Rsuch | f (x)−ℓ| <ε for all x <L,x ∈D and write limx→−∞ f (x) =ℓ. We can also define limx→∞ f (x) =±∞and limx→−∞ f (x) =±∞in a similar way. ■ Example 3.2.8 We prove from the definition that lim x→∞ 3x2 2x2 +1 = 3 2 . The approach is similar to that for sequences, with the difference that x need not be an integer. Let ε >0. We want to findMsuch that for all x >M, 3x2 2x2 +1− 3 2 <ε. (3.3) Now, 3x2 2x2 +1− 3 2 = 6x2 −6x2 −3 2(2x2 +1) = − 3 4x2 +2 = 3 4x2 +2 . To simplify the calculations it will be convenient to assume that x >0. This assumption is justified since we can always choose M>0. Since 4x2 +2>4x2, we have 3x2 2x2 +1− 3 2 = 3 4x2 +2 < 3 4x2 (3.4)

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