1. Mapping is a correspondence relationship between sets.

2. The mapping from number set to number set is a function.

3. Note: Generally speaking, mappings are single-valued mappings, that is, one independent variable can only correspond to one dependent variable, otherwise it corresponds to multiple dependent variables, which is a set-valued mapping.

4. Mapping is also called operator. According to different set situations, in different branches of mathematics, mapping has different names, such as: functional (non-empty set to number set), transformation (non-empty set to it). itself), function (set of real numbers to set of real numbers), etc.

2. Inverse mapping

1. Only bijections can define inverse mappings.

2. Inverse mapping definition: For each , there is a unique and certain , such that (that is, corresponding to it), the corresponding rule is recorded as , and its expression is .

3. Inverse mapping exists only for injectives.

4. Composite mapping definition: Given two mappings, among them, a corresponding rule from to can be defined by the mapping sum, which maps each to. Obviously, this corresponding law determines a mapping from to. This mapping is called a composite mapping composed of mapping and mapping, denoted as, that is, . Note: The value range of must be included in the definition domain of . Right now. Otherwise, a composite function cannot be formed.

1. Definition of sequence limit:

Suppose {x_n} is a sequence. If there is a constant a, for any given positive number ε (no matter how small it is), there is always a positive integer N, so that when n > N, the inequality |x_n - a| < ε If established, then the constant a is said to be the limit of the sequence {x_n}, or the sequence {x_n} converges to a. Denote it as lim x_n = a.

2. Properties of the limit of a sequence:

(1) Uniqueness: The limit of any convergent sequence {x_n} is unique.

(2) Boundedness: If the sequence {x_n} converges, then the sequence {x_n} must be bounded. That is to say, for the converged sequence {x_n}, any sub-sequence {x_nk} also converges, and the limit is the same.

3. Limits of functions:

(1) The limit of the function when x→infinity or x→X_0: Whether a function has a limit at a certain point has nothing to do with whether it is meaningful at that point. In other words, the function is meaningless at a certain point, but there may be a limit at this point.

Overview: When the number n of the items in the sequence increases infinitely, the items in the sequence are infinitely close to a certain number. This certain number is called the limit of the sequence.

The function limit is defined as follows:

Suppose the function f(x) is defined in a certain decentered neighborhood of point x0. If there is a constant A, for any given positive number ε, there will always be a positive number δ, such that when x satisfies the inequality 0 < |x - x0 | < δ, the corresponding function values ​​f(x) all satisfy the inequality |f(x) - A| < ε, then the constant A is called the limit of the function f(x) when x tends to x0, denoted as lim x →x0 f(x) = A.

1. Uniqueness: If the function f(x) has a limit at point x0, then this limit is unique.

2. Local boundedness: If the function f(x) has a limit at point x0, then there is a decentered neighborhood such that f(x) is bounded in this neighborhood.

3. Local sign preservation: If the function f(x) has a limit at point x0, and in a certain decentered neighborhood of point x0, the sign of f(x) is the same as the sign of A (that is, f(x) > 0, then A > 0), then in this neighborhood, the sign of f(x) is the same as the sign of A.

1. Infinitely small:

(1) Infinitesimal is a variable (function) whose limit is zero. That is to say, it can be infinitely close to zero in a certain change process, but it will never be equal to zero. For example, when x→x0, the limit of f(x) is 0, then f(x) is called infinitesimal when x→x0.

(2) Infinitely small is relative to a certain change process of the independent variable. For example, as x→∞, 1/x→0, so 1/x is infinitesimal as x→∞.

(3) The sum of a finite number of infinitesimals is infinitesimal, and the product of a finite number of infinitesimals is also infinitesimal. For example, (1/x) (1/x^2) (1/x^3) = (x x^2 x^3)/x^4 = 1/x^2 is infinitesimal when x→∞.

(4) The product of a bounded function and an infinitesimal is infinitesimal. For example, sin(1/x) is a bounded function. When x→0, sin(1/x) and 1/x are equivalent to infinitesimal.

(5) The product of a constant and an infinitesimal is an infinitesimal. For example, 2 * (1/x) = 2/x is infinitesimal as x→∞.

2. Infinity:

(1) A variable whose absolute value increases infinitely is called infinity. For example, when x→∞, x is positive infinity; when x→-∞, x is negative infinity.

(2) The relationship between infinity and infinitesimal: Positive infinity and negative infinity are both situations where the limit does not exist. In addition, infinity must be an unbounded function, but an unbounded function is not necessarily infinite.

(3) If f(x) has a limit of A when x tends to x0, then f(x)=A a(x), where the limit of a(x) is equal to 0 when x tends to x0. This is the relationship between infinitesimals and function limits.

1. The sum of two infinitesimals is infinitesimal.

2. The sum of a finite number of infinitesimals is also infinitesimal.

3. The product of a bounded function and an infinitesimal is infinitesimal.

4. The product of a constant and an infinitesimal is an infinitesimal.

5. The product of a finite number of infinitesimals is infinitesimal.

6. If lim f(x) = A and lim g(x) = B, then:

(1) lim [f(x) ± g(x)] = lim f(x) ± lim g(x) = A ± B.

(2) lim [f(x) * g(x)] = lim f(x) * lim g(x) = A * B.

Two important limits:

* lim x→0 sin(x) / x = 1

* lim x→∞ (1 1/x)^x = e

Several common equivalent infinitesimal substitutions (when x approaches 0):

1. sinx~x

2. arcsinx ~ x

3. tanx ~ x

4. arctanx ~ x

5. ln(1 x) ~x

6.e^x-1~x

For the function f(x), if f(x) is continuous at point x0, then there is the following definition:

Suppose the function f(x) is defined in a certain neighborhood of point x0. If lim Δx→0 Δy=lim Δx→0 [f(x0 Δx)-f(x0)]=0, then we call the function f(x ) is continuous at point x0.

This definition shows that as x gradually approaches x0, the difference between the function values ​​f(x) and f(x0) tends to 0.

In addition, we also need to understand what a discontinuity point is. If the function f(x) is discontinuous at point x0, then x0 is the discontinuity point of f(x). Discontinuity points can be divided into first type discontinuity points and second type discontinuity points.

The first type of discontinuity points includes detachable discontinuity points and jump discontinuity points. A discontinuous point means that at that point, the function is defined but the limit does not exist. A jump discontinuity means that at that point, the left and right limits of the function are not equal.

The second type of discontinuities includes infinite discontinuities and oscillating discontinuities. The point of infinite discontinuity means that at that point, the limit of the function is infinity. The oscillation discontinuity point means that at this point, the limit of the function oscillates indefinitely.