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Catalogue of Typical Problems and Methods in Mathematical Analysis
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The first chapter is the limit of unary function.

1. 1 function

First, about the inverse function.

Second, odd function, even function

Third, the periodic function

Four, several commonly used inequalities

Fifth, find the general term of recursive sequence.

1.2 Prove the existence of limit by definition

First, prove the limit by definition.

Second, prove the limit with Cauchy criterion

Third, negative form.

4. Prove the existence of limit by monotone boundedness principle.

Fifth, the limit relationship between sequence and subsequence, function and sequence.

VI. Restrictions on the operability of symbols

The first chapter is the limit of unary function.

1. 1 function

First, about the inverse function.

Second, odd function, even function

Third, the periodic function

Four, several commonly used inequalities

Fifth, find the general term of recursive sequence.

1.2 Prove the existence of limit by definition

First, prove the limit by definition.

Second, prove the limit with Cauchy criterion

Third, negative form.

4. Prove the existence of limit by monotone boundedness principle.

Fifth, the limit relationship between sequence and subsequence, function and sequence.

Sixth, the operational nature of limit.

Several methods to find the limit value in 1.3

First of all, use equivalent substitution and elementary deformation to find the limit.

A. Equivalent substitution

B. Using elementary deformation to find the limit

Second, using known limits

Third, use variable substitution to find the limit.

Fourth, the two-sided attack rule

Fifthly, the generalized form of double-sided clamping rule.

Six, other commonly used methods to find the limit

A. Hospital (usually translated as L'H?pital) rules.

B. Find the limit with Taylor formula

C. Find the limit with integral definition

D. Using series to solve limit problems

E. using continuity to find the limit

F. Comprehensive examples

1.4O.Stolz formula

First, the situation of the series

Second, the situation of functional limits

Recursive form of 1.5 limit

First, use existence to seek the limit.

Second, write the general term of limit.

Third, displacement and deformation.

Fourth, the graphic method

Generalization of fixed point method of verb (abbreviation of verb)

Sixth, the application of stolz formula.

Upper and lower bounds of 1.6 sequence

1. Describe the upper and lower limits in ε-N language.

Second, the upper and lower bounds are described by the limit of the sub-column.

Third, the upper and lower bounds are described by the limit of the supremum.

Fourthly, the upper and lower bounds are used to study the limit of sequence.

5. Operational properties of upper and lower limits

Upper and lower bounds of 1.7 function

Definition and equivalent description of upper and lower bounds of 1. function

Second, the unilateral upper and lower limits

Three. Inequalities of upper and lower bounds of functions

1.8 real number and its basic theorem

First, introduce real numbers.

Second, the basic theorem of real numbers

Chapter II Continuity of Univariate Functions

Chapter III One-dimensional Calculus

The fourth chapter is the integral of unary function.

The fifth chapter series

Chapter VI Differential of Multivariate Functions

Chapter VII Multivariate Integral calculus