General function (only tells the conditions that f(x) satisfies, no expression for f(x))

f(x) represents a specific function (if the conditions are met, the simpler the better)

1. Set the coefficients according to the decomposition form

2. Compare the original polynomial with the set polynomial to get the coefficients

1. Observe that a root of the polynomial x1

2. Polynomial/(x-x1) can get the quadratic factor of the polynomial

I. The role of ε and N

II. The geometric significance of the limit of a sequence

III. Whether the limit of the sequence exists or what the value of the limit is has nothing to do with the previous finite term of the sequence.

IV. The limit of the sequence exists<-->The limits of the odd and even subsequences exist and are equal

I. The role of ε and δ

II. The geometric meaning of function limits

III. x->x0 but x≠x0 (P9)

IV. The limit at a certain point (bilateral) exists <-> the left limit and the right limit (two unilateral) exist and are equal

The independent variable tends to infinity

The independent variable tends to a finite value

It is necessary to divide the left and right limits to find the limit

A variable with a limit of 0 (in a certain limit process)

0 is the only constant that can be infinitesimal

Study the speed at which variables tend to 0 during a certain limit process

The bridge between functions and limits

How big should the absolute value be (in a certain limit process)

Pair, power, index, factorial, power index (small->large)

When the function approaches a certain point, the limit tends to infinity --> This point is the vertical asymptote of the function graph.

The sum of a finite number of infinitesimals is still infinitesimal

The product of a finite number of infinitesimals is still infinitesimal

The product of an infinitesimal quantity and a bounded quantity is still infinitesimal

The product of finite infinities is still infinite

The inner layer has limits

The outer layer has limits

The function value of the inner function cannot be equal to u0

sequence

function

sequence

function

Can be factorized first by factoring-->Generally, non-zero constant factors can be calculated

Use the four arithmetic rules of limit-->the limit is equal to the non-zero constant factor and calculate first

1. The Limit of Polynomial Fractions (Catch the Boss)

The denominator limit is 0--->Zero factor elimination method--->Operation rule of quotient

Conditions of Use

The meaning of Taylor's formula

Basic Taylor formula

Use the pinching principle to find limits

Find the limit using the monotonic bounded criterion

Domains and Correspondence Rules

piecewise function

Composite function

Inverse function

elementary functions

Discuss on the interval I (if no specific interval is specified, discuss on the definition domain)

To discuss derivative functions (functions that are differentiable) and original functions (functions that are continuous), you need to first determine whether they exist

Strictly monotonic increasing (without equal sign) and monotonic non-decreasing (with equal sign)

determination

The domain is symmetric about the origin

If f(0) exists as an odd function, then f(0)=0

determination

Periodic function Any constant is still a periodic function

determination

Bounded, unbounded

determination

1. When the independent variable increment is 0, the corresponding function value increment is 0

2. The limit of a function exists at a certain point and the limit value = the function value at that point (three meanings)

equivalence

function limit

function continuous

The function is continuous within the interval

The function is continuous within the interval

The function is continuous at the endpoints

Discontinuity points that exist in both left and right limits

Can remove discontinuities

jump break point

There is at least one non-existent discontinuity point in the left and right limits

infinite discontinuity

Oscillation break point

other

continuous within the domain of definition

If there is a definition point, it must be continuous

Continuous within the definition interval (interval included in the definition domain)

Even if there are defined points, they are not necessarily continuous (discrete points). It needs to be judged whether they are within the defined interval.

Piecewise function breaking point

The expressions on the left and right sides of this point are different

Function has no definition point

The first type of discontinuity point needs to be explained: you can jump to OR

There is no need to specify which type of discontinuity point of the second type

1. Function has no definition

2. The expressions on the left and right sides of the click are different

3. The limits on the left and right sides of the click are different

The function is defined at this point

The function has a limit at this point

The limit value of the function at this point = function value

basic elementary functions

elementary functions