[Intensive Lecture] Analysis of the Nature of Limits in Advanced Mathematics
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Table of contents
[Intensive Lecture] Analysis of the Nature of Limits in Advanced Mathematics
Necessary memory knowledge points
Examples (used to familiarize yourself with the properties of limits in advanced mathematics)
preface
In advanced mathematics, limit is one of the important concepts in the study of sequence and function. The properties of limits describe the behavior and characteristics of sequences and functions as they approach a certain value. Understanding the properties of limits is crucial for solving limits, proving the existence of limits, and calculating limit values. This article will explain in detail the properties of limits in advanced mathematics, including basic properties, four arithmetic operations, limit existence, clamping criterion and uniqueness, etc.
1. Basic properties
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Boundedness: If a function or sequence is bounded in some interval or range, then its limit exists.
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Monotonicity: If a function or sequence increases (decreases) monotonically, then its limit exists.
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Limit of a constant: The limit of a constant is equal to itself, ie lim(c) = c, where c is a constant.
Two, four operations
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Limits of sums and differences: If two functions or sequences have limits, then the limit of their sum (difference) is equal to the sum (difference) of their limits.
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Limit of product: If two functions or sequences have limits, then the limit of their product is equal to the product of their limits.
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Limits of quotients: If two functions or sequences have limits and the limits of the divisors are not zero, then the limit of their quotients is equal to the quotient of their limits.
3. Existence of Limits
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Clamping Criterion: For a sequence or function, if there are two other sequences or functions whose limit is a certain number L and sandwiched between them, then the limit of this sequence or function is also L.
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Boundedness of convergent sequence: Convergent sequence is bounded, that is, there is an upper bound and a lower bound, so that all items of the sequence are within this range.
4. Uniqueness
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The uniqueness of the limit: The limit of the sequence or function (if it exists) is unique, that is, the limit value can only be a number.
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The limit of the composite function: If the limit of the function f(x) at x = a exists and is L, and g(x) is continuous at L, then the composite function (g ∘ f)(x) at x = a The limit is L.
5. Other properties
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Zero limit law: If the limit of one function or sequence is zero and the limit of another function or sequence is a finite number, then the product of the limits of the two functions or sequences is zero.
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Limits of infinity: If the limit of a function or sequence is positive infinity or negative infinity, then their absolute value function or sequence has a limit of positive infinity.
Necessary memory knowledge points
Examples (used to familiarize yourself with the properties of limits in advanced mathematics)
Example 1
Example 2
Example 3
Example 4
in conclusion
The properties of limits are important tools for studying the behavior of sequences and functions. Fundamental properties tell us about special cases of limits, such as boundedness and monotonicity. The Four Arithmetic Rules allow us to calculate the limits of complex expressions given the known limits of functions or sequences. The limit existence and pinch criterion help us to judge whether the limit exists or not. Uniqueness guarantees the uniqueness of the limit. Other properties such as the zero limit law and the limit of infinity extend our understanding of limits. A deep understanding of the nature of limits helps us to better deal with and apply the concept of limits in mathematical analysis and applications.