Theorem 2? The product of bounded function and infinitesimal is infinitesimal;

Inference 1? The product of constant and infinitesimal is infinitesimal; The product of finite infinitesimals is infinitesimal;

Theorem 3? If, then

( 1)

(2)

(3) If yes, then:

Inference 2? If exists and is a constant, then

Inference 3? If exists and is a positive integer, then

Theorem 4? If, and then.

And in the process of the change of the same independent variable is infinitesimal, in the process of this change is the limit, so the definition is as follows:

The necessary and sufficient condition for theorem 1 sum to be equivalent infinitesimal is

Theorem 2? Set, and exist, then

Standard 1? If connected in series, and the following conditions are met:

(1) from a project, that is, when there is

(2)

Then the limit of the sequence exists, and

Standard 1'? if

(1) When (or),

(2)

Then it exists and equals

Rule two? Monotone bounded sequence must have a limit.

Cauchy limit existence criterion? The necessary and sufficient condition for the convergence of sequence is that for any given positive integer, there exists such a positive integer that when, there exists.

Application:

Conclusion 1? The basic elementary function is continuous in its domain.

note:

(1) Advanced mathematics divides basic elementary functions into five categories: power function, exponential function, logarithmic function, trigonometric function and inverse trigonometric function.

(2) Mathematical analysis divides basic elementary functions into six categories: power function, exponential function, logarithmic function, trigonometric function, inverse trigonometric function and constant function.

Conclusion 2? All elementary functions are continuous within their defined intervals.

note:

(1) Elementary function is a function generated by the combination of basic elementary function through finite rational operations (addition, subtraction, multiplication, division, rational power and rational root) and finite function, which can be expressed by an analytical expression.

Zero theorem? Let the functions be continuous in the closed interval and have the same sign (i.e.), then at least one point in the open interval makes

Intermediate value theorem? Let the function be continuous in the closed interval and take different function values at the end of this interval, then for any number between and, there is at least one point in the open interval.

Let a function be defined in the neighborhood of a point. When the independent variable gets an increment (the point is still in the neighborhood), the corresponding function gets an increment. If the ratio of sum exists when the limit exists, the function is said to be differentiable at this point, and this limit is called the derivative of the function at this point, which is recorded as

The continuity of a function at a certain point is a necessary condition for the function to be derivable at that point, but it is not a sufficient condition.

Discrimination of extreme value? If it is set as a derivative, however, when it is an even number,

What is the identification of inflection point? Let it be the derivative of order, however, when it is odd, it is the inflection point of the curve.

Asymptote?

(1) then the function has an asymptote;

(2) The function has asymptote;

(3)

Then this function has an asymptote.

Let a function be defined in an interval. In this interval, if the increment can be expressed as a constant that does not depend on it, then the function is differentiable at one point, which is called the differential of the function at a point corresponding to the increment of the independent variable, and is recorded as

Conclusion 1? The necessary and sufficient condition for a function to be differentiable at one point is that it is differentiable at one point, and when it is differentiable at one point, its differential must be

Fermat theorem