高等数学 海涅定理 证明问题

2024年11月23日 10:15
有5个网友回答
网友(1):

证明过程如下图:




         海涅定理: lim[x->a]f(x)=b存在的充要条件是:对属于函数f(x)定义域的任意数列,且lim[n->∞]an = a,an不等于a,有lim[n->∞]f(an)=b。
海涅定理表明了函数极限与数列极限的关系。如果极限lim[x→x0]f(x)存在,{xn}为函数f(x)的定义域内任一收敛于x0的数列,且满足:xn≠x0(n∈N+),那么相应的函数值数列{f(xn)}必收敛,且lim[n→∞]f(xn)=lim[x→x0]f(x).

        海涅定理是沟通数列极限与函数极限的桥梁。根据海涅定理,求函数极限则可化为求数列极限,同样求数列极限也可转化为求函数极限。因此,函数极限的所有性 质都可用数列极限的有关性质来加以证明。根据海涅定理的必要重要条件还可以判断函数极限是否存在。所以在求数列或函数极限时,海涅定理起着重要的作用。 海涅定理是德国数学家海涅(Heine)给出的,应用海涅定理人们可把函数极限问题转化(归结)成数列问题,因而人们又称它为归结原则。
        虽然数列极限与函数极限是分别独立定义的,但是两者是有联系的.海涅定理深刻地揭示了变量变化的整体与部分、连续与离散之间的关系, 从而给数列极限与函数极限之间架起了一座可以互相沟通的桥梁.它指出函数极限可化为数列极限,反之亦然.在极限论中海涅定理处于重要地位.有了海涅定理之后,有关函数极限的定理都可借助已知相应的数列极限的定理予以证明.

网友(2):

不能完全帮你解答...不过在高数里面,很多开始的假设或者令什么等于什么都是经过计算,发现当它取某些值的时候,可以更容易得出结论

网友(3):

这是一种构造,将Xn代入,根据sin的特性,sin(nπ)在n取1,2,3,4,。。。。时只有0一个值,那么f(x)极限肯定是唯一的0。Xn取另一种取法(视频后面的内容)时,得出极限是1,利用海涅定理,证明极限不存在。Xn的取值这样取既保证了Xn的极限存在(0),同时f(x)存在且好求。可以使问题简化,这是种很常见的构造法。

网友(4):

我也在看那个视频 以我的理解是取Xn=1/(n∏)和后面取Xn'=1/(2n∏+∏/2)这两个取分数的目的是为了
1.满足之前f(x)=sin1/x的函数式。
2.让他们同时落在关于X0 的领域内,既要求他们都趋向于X0。只有在两个都为分数时才能无论n怎么取都落在一个关于X0领域内(落在一个较小的范围内)。
其实Xn=1/(n∏)和后面取的Xn’=1/(2n∏+∏/2)我们可以看作是两个函数式。海涅定理是证明把Xn=1/(n∏)和后面取的Xn’=1/(2n∏+∏/2)带入
f(x)后成为两个复合函数{ f(Xn)和f(Xn') }后,若f(Xn)和f(Xn')有相同的极限,则f(x)有极限。

这是我自己的理解 希望对你有用。

网友(5):

任意取
只有证明极限不存在就OK

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