A functional relationship is a dependency between variables.
The limit method is a basic method for studying variables .
Mapping is a fundamental concept in modern mathematics, and function is a special case of mapping.
Functions are the object of study in calculus.
Here we will introduce mappings and functions, the limits of sequences and their properties, the limits of functions and their properties, infinity and infinitesimals, the algorithm of limits, the criterion of limit existence and two important limits, the comparison of infinitesimals, the continuity and discontinuity of functions, The operation of continuous functions and the continuity of elementary functions, and the properties of continuous functions on closed intervals.
1. Mappings and functions
Definition of mapping: I won't go into details here
The definitions of surjective, single injection and one-to-one mapping (bijection) will not be repeated here.
Mappings have different idiomatic names in different branches of mathematics. A mapping from a non-empty set X to a number set Y is called a functional on X, and a mapping from a non-empty set X to itself is called a transformation on X. The mapping of the set of real numbers (or a subset thereof) X to the set of real numbers Y is called a function defined on X.
The definition of inverse mapping and compound mapping will not be repeated here
Function definition: I won't go into details here
A function is a mapping from a set (or subset) of real numbers to a set of real numbers, and its value range is always within the set of real numbers, so the elements that constitute a function are: the domain of definition and the corresponding law
The domain of a function is determined in two cases: the range of arguments that make the actual problem meaningful and the range of arguments that make the functional expression meaningful .
Three representation methods of functions: tabular method, graphical method, analytical method (formula method)
The set of points on the coordinate plane corresponding to the function is called the graph of the function
Four properties of functions: boundedness, monotonicity, parity, periodicity
Usually we refer to the period of a periodic function as the smallest positive period. Not every periodic function has a minimum positive period
Inverse function and compound function (corresponding to inverse mapping and compound mapping): the definition will not be repeated
The graphs of the direct function (original function) and the inverse function are symmetrical about the y=x line
Note the ordering of composite maps and composite functions. In order to satisfy the composition requirement, it is sometimes necessary to restrict the domain of the innermost function
Operation of functions: sum, difference, product, quotient. It is necessary to pay attention to how the range of the independent variable that makes the operation expression true is determined
It can be considered that inverse functions and composite functions are also functional operations
Basic elementary functions : exponential, logarithmic, power, trigonometric, inverse trigonometric
Elementary function : a function composed of constants and basic elementary functions through a limited number of four arithmetic operations and a limited number of functional compounding steps and can be represented by a formula (excluding the inverse function operation steps)
2. Limits and Properties of Sequences
The definition of the limit of the sequence: I won't repeat it here
The positive integer N in the definition is related to any given positive number (E), and N is selected with the given positive number
When using the definition of the limit of a sequence to prove that a certain number is the limit of a sequence, it is important to be able to point out that the positive integer N mentioned in the definition does exist for any given positive number (E).
Theorem 1: If a sequence converges, then its limit is unique
Theorem 2: If a sequence converges, then the sequence must be bounded
Theorem 3: Number preservation
Theorem 4: The relationship between a convergent sequence and its subsequences
A bounded sequence is a necessary but not sufficient condition for the sequence to converge
not converge, diverge
3. Limits of functions and their properties
In a certain change process of the independent variable, if the corresponding function value is infinitely close to a certain number, then this certain number is called the limit of the function in this change process. This limit is closely related to the change process of the independent variable .
Note the definition of neighborhood and decentered neighborhood: both are open intervals
The limit of the function as the independent variable tends to a finite value:
First assume that the function is defined in the decentered neighborhood of this finite value
Since the independent variable is only approaching the value (not equal, that's what the definition says), whether the function has a limit when it tends to the value has nothing to do with whether the function is defined at the value
Left limit and right limit: no more details
A sufficient and necessary condition for the existence of a limit of a function when the independent variable tends to a certain value is that the left limit and the right limit exist and are equal.
The limit of the function as the independent variable approaches infinity:
Similar to the limit of a sequence
Theorem 1: If a function limit exists, then this limit is unique
Theorem 2: Locally boundedness of the limit of a function
Theorem 3: Local number preservation of function limits
Theorem 3': Reinforcement of Theorem 3
inference
Theorem 4: The relationship between the limit of a function and the limit of a sequence
4. Infinite and infinitely small
The definition of infinitesimal: no more details
Zero can be the only constant that is infinitely small
Theorem 1: No more details
Definition of infinity: no more elaboration
In fact, in a certain change process of the independent variable, the limit of the function that tends to infinity does not exist, but for the convenience of describing the behavior of the function, we also say "the limit of the function is infinity"
Theorem 2: No more details
5. Limit algorithm
The Four Algorithms of Limits and the Limit Algorithms of Compound Functions
Theorem 1: The sum of two infinitesimals is infinitesimal
Finite infinitesimal sums are also infinitesimals
Theorem 2: The product of a bounded function and an infinitesimal is infinitesimal
Corollary 1: The product of a constant and an infinitesimal is infinitesimal
Corollary 2: The product of finite infinitesimals is infinitesimal
Theorem 3: No more details
Corollary 1: No more elaboration
Corollary 2: No more elaboration
Theorem 4: Similar to Theorem 2, except that it applies to sequences of numbers
Theorem 5: No more details
To find the limit of a rational whole function (polynomial) or rational fraction function when the independent variable tends to a certain value, just use the value to replace the independent variable in the function expression (for rational fraction function, it is necessary to assume that after this substitution denominator not equal to zero)
For the exception mentioned above, because the independent variable tends to the value but not equal to the value, it is possible to reduce the non-zero common factor in the process of the independent variable trend (provided that the common factor exists)
The rational fraction function has a general formula for finding the limit when the independent variable tends to infinity
Theorem 6: Thoroughly understand the theorem
6. Limit existence criterion and two important limits
Criterion I: Applied to Sequences
Criterion I': Apply to Functions
Criterion I and Criterion I' are both called pinch criteria
Criterion II: Monotonic Bounded Sequences Must Have Limits This is a Sufficient Condition Applied to Sequences
A convergent sequence must be bounded, and a bounded sequence is not necessarily convergent. Sequences are bounded and monotonically convergent
The monotonic sequence mentioned here is generalized (including the equality case), different from the function
Criterion II': applied to functions
Cauchy Limit Existence Criterion (Cauchy's Convergence Principle): Sufficient and Necessary Conditions Applied to Sequences
7. The infinitesimal comparison
The sum, difference and product of two infinitesimals are still infinitesimal. However, with respect to two infinitesimal quotients, a different situation arises
The various cases of the limit of the ratio of two infinitesimals reflect how "quickly" different infinitesimals tend to zero
Definition: not to repeat
Theorem 1: No more details
Theorem 2: I won't repeat it. When applying this theorem, it must be the same variable range of the independent variable.
8. Continuity and discontinuity of functions
Definition of function continuity: no more details
Figure out the relationship between limit and continuity
Left continuous and right continuous: no more details
A rational integer function is continuous over the interval of real numbers, and a rational fraction function is continuous at every point in its domain
Definition of function break point: one of three conditions can be satisfied
Classification of discontinuous points: If x0 is the discontinuous point of the function, but both the left and right limits at x0 exist, this discontinuous point is called the first type of discontinuous point of the function. In addition, it is the second type of discontinuity.
In the first type of discontinuity, those whose left and right limits are equal are called removable discontinuities , and those that are not equal are called jumping discontinuities . Infinite discontinuities and oscillating discontinuities are the second type of discontinuity.
9. Operation of continuous functions and continuity of elementary functions
The continuity of the sum, difference, product, and quotient of continuous functions. Note the domain of the quotient
Continuity of Inverse Functions
Theorem 3: Satisfying the theorem, the limit symbols can be swapped
Theorem 4: Continuity of Composite Functions
Basic elementary functions are continuous in their domain
All elementary functions are continuous in their definition interval. The so-called definition interval is the interval included in the definition domain
If a function is known to be continuous at a certain point, then when seeking the limit of the function at that point, only the function value of the function at that point is required.
If a function is an elementary function and a point is within its definition interval, then the limit value of the function at that point is the value of the function at that point
On the Method of Calculating the Limit of Power Exponent Function
10. Properties of Continuous Functions on Closed Intervals
Definition of a function that is continuous over a closed interval
Theorem 1 (Boundedness and Maximum Minimum Theorem): A function that is continuous on a closed interval is bounded on the interval and must be able to obtain its maximum and minimum values.
Theorem 2 (zero-point theorem): no more details
Theorem 3 (Intermediate Value Theorem): No more details
Corollary: no more elaboration
Difference Between Continuity and Consistent Continuity
If the function is consistent and continuous in a certain interval, then the function is also continuous in this interval, and vice versa is not necessarily true
Theorem 4 (Uniform Continuity Theorem): If a function is continuous on a closed interval, then it is uniformly continuous on that interval.