不定式的运算法

A. 高一不等式运算法则

不等式两边相加或相减同一个数或式子，不等号的方向不变。（移项要变号）

不等式两边相乘或相除同一个正数，不等号的方向不变。（相当系数化1，这是得正数才能使用）

不等式两边乘或除以同一个负数，不等号的方向改变。（÷或×1个负数的时候要变号）

确定解集

1.比两个值都大，就比大的还大(同大取大）；

2.比两个值都小，就比小的还小(同小取小）；

3.比大的大，比小的小，无解（大大小小取不了）；

4.比小的大，比大的小，有解在中间（小大大小取中间）。

三个或三个以上不等式组成的不等式组，可以类推。

不等式的特殊性质

不等式性质1：不等式的两边同时加上（或减去）同一个数（或式子），不等号的方向不变；

不等式性质2：不等式的两边同时乘（或除以）同一个正数，不等号的方向不变；

不等式性质3：不等式的两边同时乘（或除以）同一个负数，不等号的方向变。 总结：当两个正数的积为定值时，它们的和有最小值；当两个正数的和为定值时，它们的积有最大值。

B. 不定式计算中错位微分法是什么

有理函数部分分式分解待定系数法+留数法。

不定积分结果 结果不唯一求导验证应该能够提高凑微分的计算能力先写别问唉。

举报数字帝国GG泛滥但是是一个计算器网页。

。。

C. 动词不定式

后加动词不定式的动词有：
afford aim appear agree arrange ask be decide bother care choose come dare demand desire determine expect elect endeavor hope fail happen help hesitatelearn long mean manage offer ought plan prepare pretend promise refuse seem tend wait wish undertake
举例：
The driver failed to see the other car in time.
司机没能及时看见另一辆车。
I happen to know the answer to your question.
我碰巧知道你那道问题的答案。

后加动名词（doing）的动词有：
admit appreciate avoid complete consider delay deny detest enre enjoy escape prevent fancy finish imagine mind miss postpone practise recall resent resist resume risk suggest face include stand understand forgive keep
举例：
(1) Would you mind turning down your radio a little, please?
(2) The squirrel was lucky that it just missed being caught.

词组后接doing
admit to prefer…to be used to lead to devote oneself to object to stick to be busy look forward to (to为介词） no goodn no use It's worth… as well as
can't help It's no use /good be tired of be fond of be capable of be afraid of be proud of think of / about hold off put offkeep on insist on count on / upon
set about be successful in good at
take up give up burst out prevent … from…
Find 的特殊用法：
Find 后可用分词做宾补，或先加形式宾语，再加形容词，最后加带to 的动词不定式。find后也可带一个从句。此类动词还有get，have。
I found him lying on the ground.
I found it important to learn.
I found that to learn English is important.
典型例题:
The next morning she found the man ___ in bed，dead.
A. lying B. lie C. lay D. laying
答案：A.find的宾语后面，用分词或分词短语，起宾语补足语作用。现在分词表达主动，也表达正在进行，过去分词表达被动。

省to的动词不定式（即动词原形）
1） 情态动词 ( 除ought 外，ought to)：
2） 使役动词 let, have, make：
3） 感官动词 see, watch, look at, notice , observe, hear, listen to, smell, feel, find 等后作宾补，省略to。
注意：在被动语态中则to 不能省掉。
I saw him dance.
=He was seen to dance.
The boss made them work the whole night.
=They were made to work the whole night.
4） would rather，had better：
5） Why… / why not…：
6） help 可带to，也可不带to, help sb (to) do sth：
7） but和except：but前是动词do时，后面出现的动词用不带to的动词不定式。
8） 由and, or和than连接的两个不定式，第二个to 可以省去：
9） 通常在discover, imagine, suppose, think, understand等词后，可以省去to be：
He is supposed (to be) nice. 他应该是个好人。
举例：
He wants to move to France and marry the girl.
He wants to do nothing but go out.
典型例题
1) ---- I usually go there by train.
---- Why not ___ by boat for a change?
A. to try going B. trying to go C. to try and go D. try going
答案：D. why not 后面接不带to 的不定式，因此选D。
2） Paul doesn't have to be made ___. He always works hard.
A. learn B. to learn C. learned D. learning
答案：B. make后接不带to 的动词不定式，当其用于被动时，to 不可省略。

动名词与不定式的关系
1） 动名词与不定式的区别：
动名词表达的是： 状态，性质，心境，抽象，经常性，已发生的
不定式表达的是： 目的，结果，原因，具体，一次性，将发生的
2） 接不定式或动名词，意义相同。
3） 动名词与不定式语义不同的有11 组：
1 stop to do stop doing
stop to do 停止，中断做某事后去做另一件事。
stop doing 停止做某事。
They stop to smoke a cigarette. 他们停下来，抽了根烟。
I must stop smoking. 我必须戒烟了。
典型例题
She reached the top of the hill and stopped ___ on a big rock by the side of the path.
A. to have rested B. resting C. to rest D. rest
答案：C。由题意可知，她到了山顶，停下来在一个路边的大石头上休息。因此，应选择"stop to do sth. 停下来去做另一件事"。而不仅仅是爬山动作的终止，所以stop doing sth.不正确。

2 forget to do forget doing
forget to do 忘记要去做某事。 (未做)
forget doing 忘记做过某事。 (已做)
The light in the office is stil on. He forgot to turn it off.
办公室的灯还在亮着，它忘记关了。(没有做关灯的动作)
He forgot turning the light off.
他忘记他已经关了灯了。 ( 已做过关灯的动作)
Don't forget to come tomorrow.
别忘了明天来。 (to come动作未做)
典型例题
---- The light in the office is still on.
---- Oh，I forgot___.
A. turning it off B. turn it off
C. to turn it off D. having turned it off
答案：C。由the light is still on 可知灯亮着，即关灯的动作没有发生，因此用forget to do sth.
而forget doing sth表示灯已经关上了，而自己忘记了这一事实。此处不符合题意。

3 remember to do remember doing
remember to do 记得去做某事 (未做)
remember doing 记得做过某事 (已做)
Remember to go to the post office after school.
记着放学后去趟邮局。
Don't you remember seeing the man before?
你不记得以前见过那个人吗?

4 regret to do regret doing
regret to do 对要做的事遗憾。 (未做)
regret doing 对做过的事遗憾、后悔。 (已做)
I regret to have to do this, but I have no choice.
我很遗憾必须这样去做，我实在没有办法。
I don't regret telling her what I thought.
我不为告诉她我的想法而后悔。
典型例题
---You were brave enough to raise objections at the meeting.
---Well, now I regret ___ that.
A. to do B. to be doing C. to have done D. having done
答案：D。regret having done sth. 对已发生的事感到遗憾。regret to do sth. 对将要做的事感到遗憾。本题为对已说的话感到后悔，因此选D。

5 cease to do cease doing
cease to do 长时间，甚至永远停做某事。
cease doing 短时停止做某事，以后还会接着做。
That department has ceased to exist forever.
那个部门已不复存在。
The girls ceased chatting for a moment when their teacher passed by.
姑娘们在老师走过时，停了会聊天。

6 try to do try doing
try to do 努力，企图做某事。
try doing 试验，试着做某事。
You must try to be more careful.
你可要多加小心。
I tried gardening but didn't succeed.
我试着种果木花卉，但未成功。

7 go on to do go on doing
go on to do 做了一件事后，接着做另一件事。
go on doing 继续做原来做的事。
After he had finished his maths，he went on to do his physics.
做完数学后，他接着去做物理。
Go on doing the other exercise after you have finished this one.
作完这个练习后，接着做其他的练习。

8 be afraid to do be afraid of doing
be afraid to do 不敢，胆怯去做某事，是主观上的原因不去做，意为"怕"；
be afraid of doing 担心出现doing的状况、结果。 doing 是客观上造成的，意为"生怕，恐怕"。
She was afraid to step further in grass because she was afraid of being bitten by a snake.
她生怕被蛇咬着，而不敢在草丛中再走一步。
She was afraid to wake her husband.
她不敢去叫醒她丈夫。
She was afraid of waking her husband.
她生怕吵醒她丈夫。

9 be interested to do be interested in doing
interested to do 对做某事感兴趣，想了解某事。
interested in doing 对某种想法感兴趣，doing 通常为想法。
I shall be interested to know what happens.
我很想知道发生了什么事。 (想了解)
I'm interested in working in Switzerland. Do you have any idea about that?
我对在瑞士工作感兴趣。你想过这事吗? (一种想法)

10 mean to do mean doing
mean to do 打算、想
mean doing 意味着
I mean to go, but my father would not allow me to.
我想去，但是我父亲不肯让我去。
To raise wage means increasing purchasing power.
赠加工资意味着增加购买力。

11 begin/ start to do begin/ start doing
begin / start to do sth
begin / start doing sth.

12 感官动词+doing/to do
感官动词 see, watch, observe, notice, look at, hear, listen to, smell, taste, feel + do 表示动作的完整性，真实性；+doing 表示动作的连续性，进行性
I saw him work in the garden yesterday.
昨天我看见他在花园里干活了。(强调"我看见了"这个事实)
I saw him working in the garden yesterday.（强调"我见他正干活"这个动作）
昨天我见他正在花园里干活。
典型例题
1）They knew her very well. They had seen her ___ up from childhood.
A. grow B. grew C. was growing D. to grow
答案：A。因题意为，他们看着她长大，因此强调的是成长的过程，而非正在长的动作，因此用see sb do sth 的句型。
2）The missing boy was last seen ___ near the river.
A. playing B. to be playing C. play D. to play
答案A. 本题强调其动作，正在河边玩，应此用see sb. doing sth句型。

D. 什么是动词不定式

动词不定式的简介
动词不定式是一种非限定性动词，由to＋动词原形构成，但在有些情况下to可以省略。它还是属于动词，本身可以带宾语和状语。动词不定式在句中可以作主语、宾语、表语、定语和状语，还可用在复合结构中，而且有完成式、进行式、完成进行式和否定形式。动词不定式的被动形式除了一般形式外还有其完成式和进行式。

动词不定式的时态、语态
动词不定式可以作以上各种成分，但它毕竟是动词，所以有动词的属性
动词不定式及其短语还可以有自己的宾语、状语，虽然动词不定式在语法上没有表面上的直接主语，但它表达的意义是动作，这一动作一定由使动者发出。这一使动者我们称之为 逻辑主语，其形式如下：
主动形式 被动形式
一般式 （not） to make （not） to be made
完成式 （not） to have made （not） to have been made
进行式 （not） to be making
完成进行式 （not） to have been making
(1)语态
如果动词不定式的逻辑主语是这个不定式所表示的动作的承受者，不定式一般要用被动语态形式。如：
It's a great honour to be invited to Mary's birthday party.（不定式作主语）
It was impossible for lost time to be made up.（不定式作主语）
I wish to be sent to work in the country.（不定式作宾语）
Can you tell me which is the car to be repaired?（不定式作定语）
He went to the hospital to be examined.（不定式作状语）
在There be结构中，修饰主语的不定式可用被动，也可用主动。如：There are still many things to t ake care of (to be taken care of).但有时两种形式表达的意思不同，如：These is nothing to do now.( We have nothing to do now.) There is nothing to be done now.(We can do nothing now.)

(2)时态
1) 现在时：一般现在时表示的动词，有时与谓语动词表示的动作同时发生，有时发生在谓语动词的动作之后。
He seems to know this.
I hope to see you again. = I hope that I'll see you again. 我希望再见到你。
2) 完成时：表示的动作发生在谓语动词表示的动作之前。
I'm sorry to have given you so much trouble.
He seems to have caught a cold.
3) 进行时： 表示动作正在进行，与谓语动词表示的动作同时发生。
He seems to be eating something.
4) 完成进行时：
She is known to have been wreaking on the problem for many years.

疑问词＋不定式结构
疑问词who,what,which,when,where,whether,how后可接不定式构成不定式短语，在句中作主语、宾语、表 语等。如：
①When to leave for London has not been decided yet. （不定式在句子中做主语）
②Mr. Smith didn't know whether to leave or stay there. （不定式在句子中做宾语）
③I asked Professor Xu how to learn English well. （不定式在句子中做间接宾语）
④The question was where to get the medicine needed. （不定式在句子中表语）
以上例句中疑问词＋不定式部分，均可转换为相应的从句形式。如：①When we shall leave…③…how I
could learn…
经常在这种结构中使用的动词有：consider,decide,discover,explain,find out,forget,hear,know,lea rn,observe,understand,wonder等。

动词不定式的语法功能
一、作宾语
1) 动词+ 不定式
afford. aim. appear. agree. arrange . ask . be . decide. bother . care . choose . come. dare. demand. desire. determine . expect. elect . endeavor .hope. fail . happen . help . hesitate .learn . long . mean. manage . offer . ought. plan . prepare. pretend . promise. refuse. seem. tend. wait . wish. undertake.
举例：
The driver failed to see the other car in time.
司机没能及时看见另一辆车。
I happen to know the answer to your question.
我碰巧知道你那道问题的答案。

2）动词+不定式 ；动词+宾语+不定式
ask, beg, choose, expect , hate, help intend like, love, need prefer, prepare, promise, want, wish…
I like to keep everything tidy. 我喜欢每件东西都保持整洁。
I like you to keep everything tidy. 我喜欢你年使每件东西都保持整洁。
I want to speak to Tom. 我想和汤姆谈话。
I want you to speak to Tom. 我想让你和汤姆谈话。

3） 动词+疑问词+ to
decide, know, consider forget, learn, remember, show, understand, see, wonder, hear, find out, explain, tell
Please show us how to do that. 请演示给我们如何去做。
There are so many kinds of tape-recorders on sale that I can't make up my mind which to buy.有这么多的录音机，我都拿不定主意买哪一种。
注意:疑问词带不定式在句中作成分时，谓语动词用单数。
The question is how to put it into practice.
问题是怎样把它付诸实施。

4)以下动词后，只能跟不定式作宾语。
如：agree,ask,aim,arrange,choose,decide,demand,expect,fail ,help,hope,lean,long,manage,offer,plan,prepare,
pretend,promise,refuse,wish 等，这些词大部分可接th at引导的从句。如：
I decided to ask for my money back.
I decided that I would ask for my money back.
When our visit to the farm was over,we expected to startback on foot.
When our visit to the farm was over, we expected that wewould start back on foot.

5)当复合宾语中的宾语是不定式时，先用形式宾语it代替不定式，把不定式置于补语之后，即：主语＋动 词＋it＋补语＋to do句式。如：
We think it quite important for us to learn a foreignlanguage well.
He feels it his ty to help the poor.

二、作补语
1) 动词+宾语+不定式(to do)
advise allow appoint believe cause challenge command compel consider declare drive enable encourage find forbid force guess hire imagine impel ince inform instruct invite judge know like order permit persuade remind report request require select send state suppose tell think train trust understand urge warn
例句：
a. Father will not allow us to play on the street.
父亲不让我们在街上玩耍。
b. We believe him to be guilty.
我们相信他是有罪的。

Find 的特殊用法：
Find 后可用分词做宾补，或先加形式宾语，再加形容词，最后加带to 的动词不定式。find后也可带一个从句。此类动词还有get，have。
I found him lying on the ground.
I found it important to learn.
I found that to learn English is important.

2) to + be 的不定式结构，作补语的动词。
Acknowledge, believe, consider, think, declare(声称), discover, fancy(设想), feel find, guess, judge, imagine, know, prove, see(理解), show, suppose, take(以为), understand
We consider Tom to be one of the best students in our class.
我们认为汤姆是班上最好的学生之一。

3) to be +形容词
Seem, appear, be said, be supposed, be believed, be thought, be known, be reported, hope, wish, desire, want, plan, expect, mean…
The book is believed to be uninteresting.
人们认为这本书没什么意思。

4) there be+不定式
believe, expect, intend, like, love, mean, prefer, want, wish, undrstand
We didn't expect there to be so many people there.我们没料到会有那么多人在哪里。
有些动词需用as 短语做补语，如regard, think believe, take, consider.
We regard Tom as our best teacher. 我们认为汤姆是我们最好的老师。
Mary took him as her father . 玛丽把他当作自己的父亲。

三、作主语
动词不定式作主语时，句子的谓语动词常用单数，其位置有以下两种：
(1)把不定式置于句首。如：
To get there by bike will take us half an hour.

(2)用it作形式主语，把真正的主语不定式置于句后，常用于下列句式中。如：
①It＋be＋名词＋to do
It's our ty to take good care of the old.

②It takes sb+some time+to do
How long did it take you to finish the work?

③It＋be＋形容词＋for sb＋to do
It is difficult for us to finish writing the compositionin a quarter of an hour.

④It＋be＋形容词＋of sb＋to do
It is stupid of you to write down everything the teachersays.

⑤It seems(appears)＋形容词＋to do
It seemed impossible to save money.

在句型③中，常用表示客观情况的形容词，如：difficult,easy,hard,important,impossible,necessary 等；在句型④中，常用careless,clever,good,foolish,honest,kind,lazy,nice,right,silly,stupid,wise等 表示赞扬或批评的词。在不定式前的sb，可看作其逻辑主语。这一句式有时相当于Sb is＋形容词＋to do句式 ，如：It'skind of you to help me with my English.=You are kind to helpme with my English.

(3)举例
1） It's easy (for me) to do that.我做这事太容易了
easy, difficult, hard, important, possible, impossible, comfortable, necessary, better;
the first, the next, the last, the best, too much, too little, not enough
It's so nice to hear your voice.
听到你的声音真高兴。
It's necessary for you to lock the car when you do not use it.
当你不用车的时候，锁车是有必要的。

2) It's very kind of you to help us. 他帮助我们，他真好。
Kind, nice, stupid, rude, clever, foolish, thoughtful, thoughtless, brave, considerate(考虑周到的), silly, selfish(自私的)
例句:
It was silly of us to believe him. 我们真愚蠢，竟然相信了他。
It seemed selfish of him not to give them anything. 他不给他们任何东西，这显得太自私了。

注意：1) 其他系动词如，look，appear等也可用于此句型
2) 不定式作为句子成分时，动词用单数形式。
3) 当不定式作主语的句子中又有一个不定式作表语时，不能用It is… to…的句型
(对)To see is to believe. 百闻不如一见。
(错)It is to believe to see.

四、作表语
不定式作表语表示具体动作或将来动作；动名词作表语表示抽象的一般行为。
①To be kind to the enemy is to be cruel to the people.
②My chief purpose is to point out the difficulties ofthe matter.
③What I would suggest is to put off the meeting.
当主语和表语都是不定式时，其含义一是条件，一是结果（例①）。当主语是aim,ty,hope,idea,mista ke,plan,purpose,suggestion等为中心词的名词词组（例②）时，或以what引导的名词性分句（例③），不定 式说明主语的内容。
④Our work is serving the people.
⑤What he likes is taking a walk after supper.
⑥The story told by Mr.Wang is interesting.
④⑤句动名词作表语，与主语部分可以转换，如Serving thepeople is out work,而⑥句中是现在分词作 表语，说明主语的性质、状态，现在分词具有形容词的各种特征，另外，动名词作表语还应与进行时态区别开 来。

五、作定语
⒈不定式作定语
不定式在句中作定语，置于被修饰的名词或代词之后。如：
①The next train to arrive is from Washington.
②Have you anything to be taken to your sister?
③Do you have anything to say on the question?
④Would you please give me some paper to write on?
⑤My wish to visit France has come true at last.

不定式短语作定语和被修饰词之间表示以下关系：
(1)表示将来的动作（例①）。
(2)与被修饰词之间有动宾关系，如是不及物动词，则需加介词（例④）。
(3)与被修饰词之间有动宾关系，同时与句中其它词之间又有逻辑上的主谓关系时，尽管有被动含义，却仍 用主动语态（例③）；如只有动宾关系，而无逻辑上的主谓关系，则需用被动语态（例②）。
(4)不定式作定语时，一般可转换为定语从句，例①to arrive=that will arrive。

六、作状语
1）目的状语
To… only to (仅仅为了), in order to, so as to, so(such)… as to… (如此……以便……)
He ran so fast as to catch the first bus. 他飞快地跑以便赶上第一班车。
I come here only to say good-bye to you. 我来仅仅是向你告别。

2）作结果状语，表事先没有预料到的，要放在句子后面。
What have I said to make you angry.
He searched the room only to find nothing.

3) 表原因
I'm glad to see you.

省to 的动词不定式
1） 情态动词 ( 除ought 外，ought to)：
2） 使役动词 let, have, make：
3） 感官动词 see, watch, look at, notice , observe, hear, listen to, smell, feel, find 等后作宾补，省略to。

注意：在被动语态中则to 不能省掉。
I saw him dance.
=He was seen to dance.
The boss made them work the whole night.
=They were made to work the whole night.
4） would rather，had better：
5） Why… / why not…：
6） help 可带to，也可不带to, help sb (to) do sth：
7） but和except：but前是动词do时，后面出现的动词用不带to的动词不定式。
8） 由and, or和than连接的两个不定式，第二个to 可以省去：
9） 通常在discover, imagine, suppose, think, understand等词后，可以省去to be：
He is supposed (to be) nice. 他应该是个好人。
举例：
He wants to move to France and marry the girl.
He wants to do nothing but go out.
比较： He wants to do nothing but go out.
He wants to believe anything but to take the medicine.

动词不定式的否定式
Tell him not to shut the window…
She pretended not to see me when I passed by. 我走过的时候，她假装没看见。

不定式的特殊句型
1、不定式的特殊句型so as to
1） 表示目的；它的否定式是so as not to do。
Tom kept quiet about the accident so as not to lose his job.
汤姆对事故保持沉默是为了不丢掉他的工作。
Go in quietly so as not to wake the baby.
轻点进去，别惊醒了婴儿。
2) so kind as to ---劳驾
Would you be so kind as to tell me the time?
劳驾，现在几点了。
2、 "Why not +动词原形"表达向某人提出建议
"为什么不……?" "干吗不……?"
例如：
Why not take a holiday?
干吗不去度假?

3、It's for sb.和 It's of sb.

1）for sb. 常用于表示事物的特征特点，表示客观形式的形容词，如easy, hard, difficult, interesting, impossible等：
It's very hard for him to study two languages. 对他来说学两门外语是很难的。
2）of sb的句型一般用表示人物的性格，品德，表示主观感情或态度的形容词，如good, kind, nice, clever, foolish, right。
It's very nice of you to help me. 你来帮助我，你真是太好了。

for 与of 的辨别方法：
用介词后面的代词作主语，用介词前边的形容词作表语，造个句子。如果道理上通顺用of，不通则用for。如：
You are nice. (通顺，所以应用of)。
He is hard. (人是困难的，不通，因此应用for。)

4、不定式的特殊句型too…to…

1）too…to 太…以至于…
He is too excited to speak.
他太激动了，说不出话来。
---- Can I help you ? 需要我帮忙吗?
---- Well, I'm afraid the box is too heavy for you to carry it, but thank you all the same. 不用了。这箱子太重，恐怕你搬不动。谢谢您。
2) 如在too前有否定词，则整个句子用否定词表达肯定， too 后那个词表达一种委婉含义，意 为"不太"。
It's never too late to mend. (谚语)
改过不嫌晚。
3) 当too 前面有only, all, but时，意思是：非常… 等于very。
I'm only too pleased to be able to help you. 我非常高兴能帮助你。
He was but too eager to get home. 他非常想回家。

动词不定式的"to"与介词的"to"区别to 有两种用法：
一为不定式+动词原形；
一为介词+名词/动名词, to 在下面的用法中是第二种，即to+ 名词/动名词：
admit to承认，confess to承认，be accustomed to 习惯于，be used to 习惯于， stick to 坚持， turn to开始，着手于， devote oneself to 献身于， be devoted to 致力于， look forward to 盼望，pay attention to 注意

介词but,except,besides+to do(do)
在这种句型中，如介词前有动词do，后面应接不带to的不定式；如无do，则接to不定式，即带do不带to， 带to不带do。如：
The enemy soldiers had no choice but to give in.
On Sunday afternoon I had nothing to do but watch TV.

动词不定式与动名词区别与联系
1） 动名词与不定式的区别：
动名词表达的是： 状态，性质，心境，抽象，经常性，已发生的 不定式表达的是： 目的，结果，原因，具体，一次性，将发生的

2） 接不定式或动名词，意义相同。

3)在下列情况下，一般要用不定式：
①hate,like,love前有would(should)时，如：I'd like to have a cup of coffee.
②当谓语动词begin,continue,start等是进行式时，如：Thestudents are starting to work on the di fficult maths problem.
③begin,continue,start与know,understand等状态动词连用时，如：I soon began to understand what was happening.
⒋advise,allow,encourage,forbid,permit等动词后接动名词作宾语，或带不定式作宾语补足语。如：
Our teachers don't permit our swimming in the lake.
Our teachers don't permit us to swim in the lake.
4） 部分动词后接不定式或动名词时，意义差别较大，应根据句子语境选择使用。
①forget,remember,regret后接不定式，表示现在或未来的动作，接动名词表示动作已经发生。如：
Don't forget fo post the letter for me.
Have you forgotten meeting her in Beijing Airport?
Remember to close the windows before you leave.
I remember writing him a letter a year ago.
We regret to tell you that all of you are not invited toattend the meeting.
They regretted ordering these books from abroad.

动名词与不定式语义不同的有11 组：
1 stop to do stop doing
2 forget to do forget doing
3 remember to do remember doing
4 regret to do regret doing
5 cease to do cease doing
6 try to do try doing
7 go on to do go on doing
8 afraid to do afraid doing
9 interested to do interested doing
10 mean to do mean doing
11 begin/ start to do begin/ start doing

1 forget doing/to do
forget to do 忘记要去做某事。 (未做)
forget doing 忘记做过某事。 (已做)
The light in the office is stil on. He forgot to turn it off.
办公室的灯还在亮着，它忘记关了。(没有做关灯的动作)
He forgot turning the light off.
他忘记他已经关了灯了。 ( 已做过关灯的动作)
Don't forget to come tomorrow.
别忘了明天来。 (to come动作未做)

2 stop doing/to do
stop to do 停止，中断做某事后去做另一件事。
stop doing 停止做某事。
They stop to smoke a cigarette. 他们停下来，抽了根烟。
I must stop smoking.. 我必须戒烟了。

3 remember doing/to do
remember to do 记得去做某事 (未做)
remember doing 记得做过某事 (已做)
Remember to go to the post office after school.
记着放学后去趟邮局。
Don't you remember seeing the man before?
你不记得以前见过那个人吗?

4 regret doing/to do
regret to do 对要做的事遗憾。 (未做)
regret doing 对做过的事遗憾、后悔。 (已做)
I regret to have to do this, but I have no choice.
我很遗憾必须这样去做，我实在没有办法。
I don't regret telling her what I thought.
我不为告诉她我的想法而后悔。

5 cease doing/to do
cease to do 长时间，甚至永远停做某事。
cease doing 短时停止做某事，以后还会接着做。
That department has ceased to exist forever.
那个部门已不复存在。
The girls ceased chatting for a moment when their teacher passed by.
姑娘们在老师走过时，停了会聊天。

6 try doing/to do
try to do 努力，企图做某事。
try doing 试验，试着做某事。
You must try to be more careful.
你可要多加小心。

7 go on doing/to do
go on to do 做了一件事后，接着做另一件事。
go on doing 继续做原来做的事。
After he had finished his maths，he went on to do his physics.
做完数学后，他接着去做物理。
Go on doing the other exercise after you have finished this one.
作完这个练习后，接着做其他的练习

8 be afraid doing/to do
be afraid to do 不敢，胆怯去做某事，是主观上的原因不去做，意为"怕"；
be afraid of doing 担心出现doing的状况、结果。 doing 是客观上造成的，意为"生怕，恐怕"。
She was afraid to step further in grass because she was afraid of being
bitten by a snake.
她生怕被蛇咬着，而不敢在草丛中再走一步。

.9 be interested doing/to do
interested to do 对做某事感兴趣，想了解某事。
interested in doing 对某种想法感兴趣，doing 通常为想法。
I shall be interested to know what happens.
我很想知道发生了什么事。 (想了解)
I'm interested in working in Switzerland. Do you have any idea about that?
我对在瑞士工作感兴趣。你想过这事吗? (一种想法)

10 mean to doing/to do
mean to do 打算、想
mean doing 意味着
I mean to go, but my father would not allow me to.
我想去，但是我父亲不肯让我去。
To raise wage means increasing purchasing power.
赠加工资意味着增加购买力。

11 begin(start) doing/to do
begin / start to do sth
begin / start doing sth.
1) 谈及一项长期活动或开始一种习惯时，使用doing.
How old were you when you first started playing the piano?
你几岁时开始弹钢琴？
2) begin, start用进行时时，后面动词用不定式to do
I was beginning to get angry。
我开始生起气来。
3) 在attempt, intend, begin, start 后接know, understand, realize这类动词时，常用不定式to
do。
I begin to understand the truth。
我开始明白真相。
4) 物作主语时
It began to melt.

12 感官动词 + doing/to do
感官动词 see, watch, observe, notice, look at, hear, listen to, smell, taste,
feel + do 表示动作的完整性，真实性；+doing 表示动作的连续性，进行性
I saw him work in the garden yesterday.
昨天我看见他在花园里干活了。(强调"我看见了"这个事实)
I saw him working in the garden yesterday.（强调"我见他正干活"这个动作）
昨天我见他正在花园里干活。

E. 不等式的运算法则

“不等式
运算法则
为不等式两边相加或相减同一个数或式子，
不等号
的方向不变。不等式两边相乘或相除同一个
正数
，不等号的方向不变。不等式两边乘或除以同一个负数，不等号的方向改变。”

F. 洛必达法则其他不定式

摘要 洛必达法则是在一定条件下通过分子分母分别求导再求极限来确定未定式值的方法 [1] 。众所周知，两个无穷小之比或两个无穷大之比的极限可能存在，也可能不存在。因此，求这类极限时往往需要适当的变形，转化成可利用极限运算法则或重要极限的形式进行计算。洛必达法则便是应用于这类极限计算的通用方法 [2] 。

G. 不等式3个运算法则如何正确使用

不等式有三种：
（1）基本不等式 设a>b则

• ac>bc(c>0)；

• ac<bc(c<0)；

• a/c>b/c(c>0)；

• a/c<b/c(c<0)；

• a^n>b^n（a>0，b>0，n>0)

（2）绝对不等式

（3）绝对值不等式

• |A+B|≤|A|+|B|；

• |A-B|≤|A|+|B|；

• |A-B|≥|A|-|B|；

• -|A|≤A≤|A|；

• √(A²)=|A|；

• |AB|=|A||B|；

• |A/B|=|A|/|B|
  
  

H. 洛比达法则适用于0/0、∞/∞型不定式，等价无穷小量不适用于不定式，请问啥是不定式啊

能用极限运算法则(四则运算，复合合函数求极限法则等)直接计算的可以叫定式，相对的，不能直接通过极限运算法则计算的就叫不定式,主要有以下形式：

∞-∞,∞*0,0/0,∞/∞,1^∞,0^0,∞^0.

计算上往往把它们化为0/0,∞/∞这两种类型，以期利用洛必达法则。