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Limit Laws
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Limit Laws
Suppose that c is a constant and and . Then
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(1).(Sum Law) .
(2).(Scalar Multiple Law) .
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Remark :
From (1) and (2), we get .
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(3).(Product Law) .
If we use the Product Law repeatedly with , we obtain the following Law.
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where n is a positive integer.
(4). If , then .
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Remarks :
1. From (3) and (4), we get
(Quotient Law) , if .
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2. Note that the condition implies that g( x ) is bounded away from 0 for x near a . In fact, there exists a positive number d such that whenever and . Hence, is well-defined for near a .
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Suppose that , what can we say about the behavior of g( x ) near a ?
Intuitively, implies that can be as small as we want by taking x sufficiently close to a (but not equal to a ). Hence, can be as large as we want by taking x sufficiently close to a (but not equal to a ). We would conjecture that . Here is the proof :
Given M > 0, since , there exists , such that whenever 0 < .We get,
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whenever 0 < .
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Hence, .
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Questions :
1. Suppose that both and exist, does exist ?
2. Suppose that both and exist, does exist ?
3. Suppose that exists and , does exist ? If so, what is the limit ?
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4. Suppose that c is a constant and and .
What can we say about the following limits ?
, , and .
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5. Suppose that and . What can we say about the following limits ?
, , and .
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Theorem If when x is near a (except possibly at a ) and both and exist, then
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.
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Proof : Let and . Suppose that .
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Taking , we have that there exists and such that
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whenever 0 < ,
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and
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whenever 0 < .
Let , if 0 < then 0 < and 0 < , so we get
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< .
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But this contradicts . Thus, the inequality must be false. Therefore, .
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Question : Suppose that when x is near a (except possibly at a ) and both and exist, is it true that
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The Squeeze Theorem
If and when x is near a (except possibly at a ) and
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=
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then .
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Proof : Given , since = , there exists and such that
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whenever 0 < ,
and
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whenever 0 < .
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Let , if 0 < then 0 < and 0 < , so we get
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< and < .
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Therefore, whenever 0 < .
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Example Show that .
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Solution :
Note that since does not exist, we can not use the Product Law.
Since
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for all
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we have , as illustrated above,
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and for all .
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It is easy to show that and . By the Squeeze Theorem, we get .
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Question :
1. Suppose that , is it true that ?
2. Suppose that exists, is it true that also exists ?
3. Suppose that , is it true that ?
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