Chapter 1 Limits and__ Continuity

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Definition 2.7 Let f be a function defined on an open interval of the form (c, a) for some real number c, and let L be a real number. We say that the left-hand limit of f(x) as x approaches a from the left is L, and write lim f ( x ) = L, lim+ f ( x) = L,
lim f ( x ) = 3 2 ,
2 1 -2 -1 1 2
and say that the limit of f(x), as x approaches 1, is 3/2, or that f(x) approaches 3/2 as x approaches 1. Figure 6
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x →1
x→a
if for every number ε>0, there is a number δ>0 such that
if 0 < x a < δ , then f ( x ) L < ε . y y=L+ε L y=L-ε o a-δ a a-δ x Figure 5
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In order to understand the precise meaning of a function in Definition 2.5, let us begin to consider the behavior of a function 0.5 x + 1, x ≠ 1 f ( x) = x =1 0, as x approaches 1. From the graph of f shown in Figure 6, we can intuitively see that as x gets closer to 1 from both sides but x≠1, f(x) gets closer to 3/2. In this case, we use the notation
As an example, we have f(x)=x2cosx>0 for every x in open interval (-1,+1), except x≠0, and lim x 2 cos x = 0 . x →0 2.2.2 Limits Involving Infinity
Now let us discussed functions whose value approaches some number L as x becomes very large, very small, or absolute value of x gets very large.
Chapter 2 Limits and Continuity 2.1 Limits of Sequences 2.1.1 Definition We usually write a sequence ｛an｝as a1, a2, a3, …,an, … . a1 is called the first term of the sequence. a2 is called the second term of the sequence. an is the general term of the nth term of sequence. For instance, ｛1/n｝, ｛(-1)n｝and｛(-1)n/n｝are all sequences. A sequence can be pictured by plotting its terms on a horizontal axis.The graph of the successive elements of the sequence ｛(-1)n/n｝is illustrated as follows. a1 a3 a5 a4 a2 -1 0 1 x Figure 4
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2.2 Limits of Functions 2.2.1 A Limit of a Function f(x) as x Tends to a Real Number x0 Definition 2.5 Let f be a function defined on some open interval containing a except possibly at a itself, and let L be a real number. We say that the limit of f(x) as x approaches a is L, and write lim f ( x ) = L,
n→∞ n n
lim If n→∞ an exist, we say the sequence converges (or is convergent). Otherwise we say the sequence diverges (or is divergent).
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Example 1 Determine whether the sequence ｛(-1)n｝is convergent or divergent. Solution If we write out the terms of the sequence, we obtain ｛-1, 1, -1, 1, -1, 1, ……｝. Since the terms oscillate between 1 and –1 infinitely often, an does not approach any number. Thus lim( 1)n n→∞ n｝is divergent. does not exist; that is , the sequence ｛(-1) Example 2 n + ( 1)n lim = 1. n→∞ n Right?
Notice that in Theorem 2.10, we require f(x) ≥0 and get the fact L≥0. In fact, the symbol “=” in the conclusion L≥0 can not be omitted although the condition f(x)>0 holds. Readers can give some counter examples in which f(x)>0, but the limit of f(x) as x approaches a is 0.
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lim can see From the above graph,→∞ an = 0 that as n gets greater we n and greater, an=(-1)n/n gets closer and closer to the origin 0. In this case, we use the notation , and we say that the limit of the sequence｛an｝ is 0 or ｛an｝ converges to 0. overlook: Definition 2.1 A sequence ｛an｝has the limit L and we write lim a = L or a → L as n → ∞,
x→a
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x x x For instance, since lim = 1 ≠ lim = 1 therefore lim , + x→0 x x→0 x x→0 x does not exist. Example 3 Suppose that 1 2 x , x > 0, f ( x) = 2 x + 1, x ≤ 0. (1) Find lim f ( x ) and lim+ f ( x ). x →0 x →0 lim f ( x ). (2) Discuss x →0 Solution (1) lim f (x) = 1 , lim f (x) = 0 .
x→a
For example, Theorem 2.8 if
x→a
a δ < x < a
then
f ( x) L < ε .
x x lim+ = 1, lim = 1. x →0 x x →0 x lim f ( x ) = L
x→a
if and only if
x→a
lim+ f ( x ) = lim f ( x ) = L.
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Theorem 2.2 If a sequence has a limit, the limit is unique. A sequence ｛an｝is an increasing sequence if an≤an+1 for every positive integer n. A sequence ｛an｝is a decreasing sequence if an≥an+1 for every positive integer n. If there exist two number a and b such that a<an<b for every n∈N, ｛an｝is called a bounded sequence. The next theorem concerns with existence of limit. Theorem 2.3 Every bounded monotonic sequence has a limit. This theorem provides only a sufficient condition for existence of limit. It is not a necessary condition. However, we do has the following nice result in the reverse direction. Theorem 2.4 Every convergent sequence is bounded.
x→0 x→0+
(2) Because lim f (x) = 1 ≠ lim f (x) = 0, so x→0 x→0+ does not exist.
.9 If lim f (x) = L > 0(<0), then f(x)>0 (<0) for x→a every x in (a-δ, a+δ), except possibly x=a. Theorem 2.10 If lim f (x) = L and f(x)≥0 (≤0) for every x in x→a some open interval (a-δ, a+δ), except possibly x=a, then L≥0（≤0）.
x → +∞
lim
f (x) = L,
1 x lim ( ) = 0. x → +∞ 2
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2.2.2 Limits Involving Infinity Now let us discussed functions whose value approaches some number L as x becomes very large, very small, or absolute value of x gets very large. Definition 2.11 Let f be a function defined on an infinite interval (c, +∞）for a real number c, and let L be a real number. We say that the limit of f(x) as x approaches +∞ is L, and write