MindMap Gallery high numerical limit
This is a mind map about advanced numerical limits. The main contents include: problem solving methods, existence of limits, sequence, functions, and properties. The detailed introduction allows you to understand and learn more quickly and conveniently. If you need it, collect it quickly!
Edited at 2024-11-28 20:31:39CBT cognitive behavioral therapy, cognitive therapy, psychological counseling, CBT basic concept: ideas determine emotions, experience determines ideas, experience requires comparison to be meaningful, and there are individual differences in experience.
Psychological perception, perception is generated on the basis of sensation. It is the response of the human brain to the objective things and overall attributes that directly act on the sensory organs. The introduction is detailed, students in need can save it.
心理學知覺,知覺在感覺的基礎上產生它是人腦對直接作用於感覺器官的客觀事物,整體屬性的反應。介紹詳細,有需要的同學,可以收藏喲。
CBT cognitive behavioral therapy, cognitive therapy, psychological counseling, CBT basic concept: ideas determine emotions, experience determines ideas, experience requires comparison to be meaningful, and there are individual differences in experience.
Psychological perception, perception is generated on the basis of sensation. It is the response of the human brain to the objective things and overall attributes that directly act on the sensory organs. The introduction is detailed, students in need can save it.
心理學知覺,知覺在感覺的基礎上產生它是人腦對直接作用於感覺器官的客觀事物,整體屬性的反應。介紹詳細,有需要的同學,可以收藏喲。
high numerical limit
Sequence
First, clarify the definition of a sequence: a function whose independent variable takes a positive integer is called a sequence (that is, a sequence is a special function)
e.g.:
The concept of sequence limits
Y=
Y=
Their limits are all 0 (it can be known from their function graph that the Y value approaches 0 infinitely)
type
diverge
There are multiple limit values or the image has no limit
convergence
There is only one clear limit
function
definition
Suppose f(X) is defined in a certain decentered neighborhood of X0
As people say, the limit is 0
Then the constant A is called the function f(X) when the limit of X—X0
category
left limit
from left to right
right limit
right to left
Asymptote
level
The limit of the function is him (set as a) y=a
vertical
As long as one side is infinity, whether it is positive infinity or negative infinity
oblique asymptote
infinite problem
The premise is that there are limits
The relationship between infinity and infinitesimal
infinity
definition
It does not mean that a large number refers to a state
infinitesimal
definition
The limit value of the function is 0
Infinitesimal comparison (compare the speed to 0 between functions)
Do business first limA/B
0
A is the higher order infinitesimal of B
endless
A is the lower order infinitesimal of B
C (constant)
A is the infinitesimal equivalent of B
special
2 sinx/x
Continuity and discontinuity
continuity
f(x) is defined in a certain decentralized region of X0
Increment (amount of change)
continuous function
Four arithmetic operations and composition between functions
concept
discontinuity
The total requirement is that f(x) is defined in the decentered domain of X0, that is, f(x) has a value.
Discontinuities of the first kind
Can remove discontinuities
Condition: F(X0^)=F(X0^-)
jump break point
Condition: F(X0^) is not equal to F(X0^-)
Type II discontinuities
infinite discontinuity
Oscillation break point
other
neither discontinuous nor continuous
Two major blocks (judgment means)
piecewise function
dividing point
elementary functions
undefined point
Properties of continuous functions on closed intervals
Maximum value theorem
A function that is continuous on a closed interval
minimum value
maximum value
Intermediate Value Theorem (Zero Point Theorem)
Use derivatives to determine monotonicity, and then algebra to the function to determine the zero point position.
F(X) is monotonically continuous on (a,b). If f(a).*f(b)<0, there is an H point H belonging to (a,b) such that f(H)=0
Notice
Functions only exist and do not exist, not divergence and convergence
nature
uniqueness
The limit value only exists (provided that the limit of the sequence exists)
local boundedness
If the sequence converges, the sequence must be bounded, and the convergence is not necessarily monotonic.
Number retention
If it is monotonic or bounded, the sequence must converge
monotonic boundedness
Monotonically increasing or decreasing, it must converge if there is an upper or lower bound
Relationship to subcolumn
A convergent sequence and its any subsequence converge to the same limit
Limit exists
Clamping Criterion
monotonic bounded criterion
How to do the questions
Two important limits
extreme type
k.*0
k/0
infinite/infinite
The way to catch the big head (find the highest power)
0/0
Approximately remove common 0 factors
Equivalent infinitesimal substitution
How to do the questions (must memorize)
Note lim (1 1/X) =e irreplaceable X-infinity
infinity-infinity
The common part is converted into 0/0 or infinity/infinity
Infinity.*0
Lópida's Law
If it’s not the infinitive, you can’t use Robida
infinitive
0/0
Conditions of use
Note: f(x) is the numerator and F(x) is the denominator
Can be converted to
can be transformed into
or this 0/0
method
You can simplify it first and then use Lupida
Derive the numerator and denominator respectively (to ensure that the position remains unchanged)
Conditions of use
Can be converted to
become
How to use
Same as 0/0
rather special
Using logarithmic identities
In essence, it is still exchanged for 0/0 or
Same as above
Same as above