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If by selecting a small punctuated neighborhood of , a variable determined by can close enough to a fix value in any level, then that variable is converge to that fix value.

Definition of Limits

Limits of Sequence

is saying:

Limits of function

the limit of of as approaches

it means that the value of the function can be made arbitrary close to by choosing x close to

is saying:

Heine's Theorem

人话：所有路径都收敛于一个值，那么就在该区域无死角收敛了。

证明：

如果函数收敛，自然每一条路径都收敛
如果函数不收敛，那么能够构造不收敛的坏数列

Sign-Preserving Property

If a function has a limit on , then there's a punctured neighborhood of , one that set, keeps the same sign with the limit

Suppose

Set , then , ,

So as when

Squeeze Theorem

if , and and has same limitation on , then also has limitation on

Order Property of Limits

If , and , then

if , then for , :

which contradicts with . So

Local Boundedness

Local Boundness

If , is bounded in some punctured neighborhood of

Rules with Real Number Operations

Constant Rule

swappable to scalar product

Proof

with , , thus:

Sum Rule

distributive to function addition

Proof: (triangular Inequality)

with ,
with ,
Thus

Product Rule

distributive to function product

Proof:

involve a trick of constructing and
be aware that is locally bounded due to the Local Boundness

:

with , , thus
with ,

so, with ,

Quotient Rule

distributive to function devision

Proof

key steps:

step1: Proof

step2: use distributive to function product:

Detail Proof

This proof is divided into two steps:

proof and reverse triangular inequality
product rule

with , , which saids:

with ,

Thus:

applying the product rule:

Composition Rule

composition rule

if is continuous, then