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MindMap Gallery Chapter 1 Function Limit Continuous Mind Map

Chapter 1 Function Limit Continuous Mind Map

Functions, limits and continuity are important concepts in calculus, and there is a close relationship between them. The limit of a function describes the value of the function at a certain point, while continuity describes the differentiability and integrability of the function at that point.

Edited at 2023-11-11 09:23:12

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Chapter 1 Function Limit Continuous Mind Map

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Outline

Advanced Mathematics Chapter 1 Limit
- 39
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Advanced Mathematics Chapter 1 Functions and Limits Mind Map
- 40
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line in landline
- 6
Darren James Bealing (Puggo)

Chapter 1 Limits and Continuity

function

The concept of function and common functions

function definition

Composite function

Inverse function

basic elementary functions

Power function

exponential function

Logarithmic function

Trigonometric functions

inverse trigonometric function

Function behavior

Monotonicity

parity

cyclical

Boundedness

limit

definition

sequence limit

limit of function

tends to a finite value

tending towards infinity

Limiting properties

General properties

uniqueness

Number retention

Boundedness

sequence limit

function limit

The relationship between limit value and infinitesimal

limit existence criterion

pinch theorem

A monotonic bounded sequence must have a limit

extreme operational properties

Four properties of arithmetic operations

Two important limits

infinitesimal

definition

infinitesimal comparison

Higher order infinitesimal

Lower order infinitesimal

Equivalent to infinitesimal

infinitesimal property

General properties

linear properties

The product of bounded and infinitesimals is still infinitesimal

The limit is represented by the function infinitesimal

Equivalence property

gigantic

definition

Infinite comparison

The relationship between infinity and unbounded variables

The relationship between infinity and infinitesimal

Continuous and discontinuous

definition

function continuous type

The function is continuous at a certain point

The function is continuous in a closed interval

function break point

Discontinuities of the first kind

Can remove discontinuities

jump break point

Type II discontinuities

infinite discontinuity

Oscillation break point

Properties of continuous functions on closed intervals

Maximum value theorem

bounded theorem

zero point theorem

medium theorem

Basic question types

I. function

Question Type 1: Composite Function

Question Type 2: Functional Behavior

II. limit

Question Type 1: The concept, nature and existence criteria of limits

concept

limit existence criterion

pinch theorem

A monotonic bounded sequence must have a limit

Question Type 2: Finding the Limit

Common methods for finding limits

Use rational arithmetic

Use basic limits to find limits

Find the limit using equivalent infinitesimal substitutions

Commonly used equivalent infinitesimal

The principle of equivalent infinitesimal substitution

Find the limit using Lópida's law

Find the limit using Taylor's formula

Use the pinch theorem to find limits

Find the limit using the definition of definite integral

Find the limit using the monotonic bounded criterion

Common question types for finding the limit

limit of function

Type 0/0

Equivalent to infinitesimal

Lópida's Law

Taylor formula

∞/∞ type

Lópida's Law

The numerator and denominator are simultaneously divided by the highest-order infinity in the numerator and denominator

∞-∞ type

Pass divided into 0/0

rationalization of radicals

Add infinite factors, then equivalent substitution or equivalent substitution, Taylor formula

0·∞ type

Transform into 0/0 type or ∞/∞ type

Type 1⁸

Make up the basic limit/(1 x)¹ⁱˣ=e

rewritten as exponent

three-part method

∞⁰ or 0⁰

Rewrite it in exponential form and convert it into 0·∞ type

The limit of a sequence

infinitive

Limit of sequence for sum of n terms

pinch theorem

Definite integral definition

summation of series

The limit of the sequence of consecutive multiplications of n terms

pinch theorem

Take the logarithm and convert it into the sum of n terms

Sequence defined by recurrence relation

Question Type 3: Determine the parameters in the limit expression

Question Type 4: Comparison of infinitesimal magnitudes

Become type 0/0

Equivalent to infinitesimal

Lópida's Law

Taylor formula

III. continuous

Question Type 1: Discuss continuity and discontinuity types

Question Type 2: Proofs of the Intermediate Value Theorem, Maximum Value Theorem and Zero Point Theorem