二元函数极限证明完整版.docx

1、二元函数极限证明完整版二元函数极限证明二元函数极限证明第一篇：二元函数极限证明二元函数极限证明设p=f教学目的：掌握二元函数的极限的定义，了解重极限与累次极限的区别与联系教学内容：二元函数的极限的定义；累次极限基本要求：较高要求：掌握重极限与累次极限的区别与联系，能用来处理极限存在性问题教学建议：要求学生弄清一元函数极限与多元函数极限的联系与区别，教会他们求多元函数极限的方法对较好学生讲清重极限与累次极限的区别与联系，通过举例介绍判别极限存在性的较完整的方法一二元函数的极限先回忆一下一元函数的极限：limf时，f法则。类似地, 二元函数基本未定型的极限问题也有相似的洛泌达法则。为了叙述上的方便

2、, 对它的特殊情形= ) 作出如下研究, 并得到相应的法则与定理 。二元函数的极限是反映函数在某一领域内的重要属性的 一个基本概念, 它刻划了当自变量趋向于某一个定值时, 函数值的变化趋势。是高等数学中一个极其重要的问题。但是, 一般来说, 二元函数的极限比起一元函数的极限, 无论从计算还是证明都具有更大的难度。本文就二元函数极限的问题作如下探讨。第四篇：二元函数的极限与连续3 二元函数的极限与连续定义设二元函数有意义, 若存在常数a,都有则称a是函数当点 趋于点或或趋于点时的极限,记作。的方式无关,即不,当（即）时，在点的某邻域内或必须注意这个极限值与点论p以什么方向和路径（也可是跳跃式地，

3、忽上忽下地）趋向分接近, 就能 使。只要p与 充与a 接近到预先任意指定的程度。注意：点p趋于点点方式可有无穷多种,比一元函数仅有左,右两个单侧极限要复杂的多（图8-7）。图8-7同样我们可用归结原则,若发现点p按两个特殊的路径趋于点时,极限在该点存在,但不相等, 则可以判定元函数极限不 存在的重要方法之一。极限不存在。这是判断多一元函数极限中除了单调有界定理外,其余的有关性质和结论, 在二元函数极限理论中都适用，在这里就不一一赘述了。例如若有, 其中。求多元函数的极限, 一般都是转化为一元函数的极限来求, 或利用夹逼定理来计算。例4 求。解由于,而,根据夹逼定理知,所以 。a0)。解例求（。

4、例6 求。解由于理知且,所以根据夹逼定.例7研究函数在点处极限是否存在。解当x2+20时，我们研究函数，沿x0,=kx0这一方式趋于（0，0）的极限，有值，可得到不同的极 限值,所以极限不存在,但,。很显然，对于不同的k。注意：极限方式的的区别, 前面两个求本质是两次求一元函数的极限, 我们称为累次极限, 而最后一个是求二元函数的极限,我们称为求二重极限。例8设函数极限都不存在,因为对任何,当时,。它关于原点的两个累次的第二项不存在极限；同理对任何时, 的第 一项也不存在极限,但是因此。由例7知, 两次累次极限存在, 但二重极限不存在。由例8可知,二重极限存在,但二个累次极限不存在。我们有下面

5、的结果:定理1 若累次极限都存在,则三者相等（证明略）。推论若但不相等,则二重极限不存在和二重极限,由于,存在。定义 设在点的某邻域内有意义,且称函数,则在点处连续,记上式称为函数的全增量。则。定义增量。为函数对x的偏二元函数连续的定义可写为偏增量。若断点, 若在点为函数（值）对的处不连续,则称点是的间在某区域在区域g上连续。若在闭区域gg上每一点都连续,则称的每一内点都连 续,并在g的连界点处成立,则称为连续曲面。在闭域g上连续。闭域上连续的二元函数的图形称关于一元函数连续的有关性质, 如最值定理、介值定理、antor定理,对于二元函数也相应成立。可以证明如下的重要结果：定理2设在平面有界闭

6、区域g上连续,则（1） 必在g上取到最大值,最小值及其中间的一切值;（2）,当时,都有。以上关于二元函数的在g上一致连续,即极限和连续的有关性质和结论在n元函数中仍然成立。第五篇：函数极限的证明函数极限的证明时函数的极限：以时和为例引入.介绍符号:的意义,的直观意义.定义几何意义介绍邻域其中为充分大的正数.然后用这些邻域语言介绍几何意义.例1验证例2验证例3验证证时函数的极限：由考虑时的极限引入.定义函数极限的“”定义.几何意义.用定义验证函数极限的基本思路.例4验证例5验证例6验证证由=为使需有为使需有于是,倘限制,就有例7验证例8验证单侧极限:1.定义：单侧极限的定义及记法.几何意义:介绍

7、半邻域然后介绍等的几何意义.例9验证证考虑使的单侧极限与双侧极限的关系:th类似有:例10证明:极限不存在.例11设函数在点的某邻域内单调.若存在,则有=2函数极限的性质教学目的：使学生掌握函数极限的基本性质。教学要求：掌握函数极限的基本性质：唯一性、局部保号性、不等式性质以及有理运算性等。教学重点：函数极限的性质及其计算。教学难点：函数极限性质证明及其应用。教学方法：讲练结合。一、组织教学：我们引进了六种极限:,.以下以极限为例讨论性质.均给出证明或简证.二、讲授新课：函数极限的性质:以下性质均以定理形式给出.1.唯一性:局部有界性:3.局部保号性:4.单调性:th4若和都存在,且存在点的空

8、心邻域,使，都有证设=註:若在th4的条件中,改“”为“”,未必就有以举例说明.5.迫敛性:6.四则运算性质:利用极限性质求极限：已证明过以下几个极限：这些极限可作为公式用.在计算一些简单极限时,有五组基本极限作为公式用,我们将陆续证明这些公式.利用极限性质，特别是运算性质求极限的原理是：通过有关性质,把所求极限化为基本极限,代入基本极限的值,即计算得所求极限.例1例2例3註:关于的有理分式当时的极限.例4例5例6例7附送：二分钟英语演讲稿二分钟英语演讲稿, hello, thank ou frequentl mention, let the material on this it ith t

9、he ne ork parable to the spiritual ivilization. eduate our students to exiting 201X, illing to shanghais future b opies of poer, if onl planted a small tree, partiipants in the shanghai orld expo foreign guests feel: this is a forest it, pollution-free it, anient and ivilized it. better it, better l

10、ife! until the time of the 201X expo, then, e ant to bee a glorious volunteers, to hanging our foreign friends in shanghai and hinas long histor, about our eduation for the orld expo ill ontribute their efforts.e expet the majorit of teahers ill be great enthusiasm for the orld expo into pratial ati

11、on to meet the orld expo, the expo ill be a total groth, and the it of development, the development of shanghai devote their efforts to advane hand in hand ith shanghai.第三篇：2分钟英语演讲稿i enter tile universit after ears of hard stud and preparation. but life in the universit is not as satisfator（满意） as h

12、at i had expeted. i bee laz and dont ant to stud.i bee silent. i bee puXXled. i dont kno hat i an do in the future. then i bee unhapp.four ears in the universit is onl a short period hen pared m hole lifetime. no it has passed a half. in this ear, man people ,suh as m parents,m friends, ask me hat i

13、 ant to do and tell me to map out a plan for m life. i dont ant to follo their suggestion, and i ant to go m . so i think arefull. i have been a oung volunteer（志愿者） for five ears.its ver happ and signifiant. then i have a dream.i ant to join the universit student volunteers go est programe. i think

14、i an be a teaher in the est.i ant to tr m best to help them and help me. i ant to see the orld earl. no i ant reah its demand and its ver diffult,but i ill ork hard in the next to ears. there is an old saing here there is a ill,there is a a.（有志者事竟成） i think m dream an e ture.no in the universit i ma

15、ture,and in the universit i prepare for the real orld.at last,i ant to sa to everbod hlod fast to our dreams,no matter ho big or small the are.the path to dreams ma not be smooth（平滑） and ide,even some sarifies（牺牲） are needed.but hold on to the end,ou an find there is no geater happiness than making

16、our dream e ture. 第四篇：3分钟英语演讲稿hi,m topi toda is: ord that has hanged the orlda ord that has hanged the orld eduation of their hildren is highl valued b parents. to get eduation in the universities abroad has been the dreams for both the parents and their hildren. to realize these dreams, the parents

17、 have to save a large amount of mone for the studies, beause international flights ost a lot. in addition, the tuition abroad is not heap. parents should also have to pa for the food, room and utilities for their hildren at the shool abroad. the hildren also need some mone for the entertainment and

18、soial events. the lothing ill ost quite a little. besides, the need extra mone to over some unexpeted expenses, like long-distane alls and so on.all in all, the parents should hek out ho muh the living and tuition ill ost for their hildren to stud abroad. the should get prepared in advane.thank ou .

19、第五篇：英语演讲稿3分钟as everone knos,english is ver important toda.it has been used everhere in the orld.it has bee the most mon language on internet and for international trade. if e an speak english ell,e ill have more hane to sueed.beause more and more people have taken notie of it,the number of the peopl

20、e ho go to learn english has inreased at a high speed.but for mself,i learn english not onl beause of its importane and its usefulness,but also beause of m love for it.hen i learn english, i an feel a different a of thinking hih gives me more room to touh the orld.hen i read english novels,i an feel

21、 the pleasure from the book hih is different from reading the translation.hen i speak english, i an feel the onfident from m ords.hen i rite english,i an see the beaut hih is not the same as our hinesei love english,it gives me a olorful dream.i hope i an travel around the orld one da. ith m good en

22、glish, i an make friends ith man people from different ontries.i an see man plaes of great intrests.i dream that i an go to london,beause it is the birth plae of english.i also ant to use m good english to introdue our great plaes to the english spoken people,i hope that the an love our ountr like us.i kno, rome as not built in a da. i believe that after ontinuous hard stud, one da i an speak english ver ell.if ou ant to be loved, ou should learn to love and be lovable. so i believe as i love english everda , it ill love me too.i am sure that i ill realize m dream one da!访问其他