多元函数极限的一种求法  

One Method of Requesting the Limit of Function of Many Variables

作　　者：丁殿坤[1] 吕端良[1] 李淑英[1] 

机构地区：[1]山东科技大学公共课部,山东泰安271019

出　　处：《南阳师范学院学报》2004年第12期25-27,共3页Journal of Nanyang Normal University

摘　　要：把多元函数极限的判断及求法转化为一元函数极限的判断及求法。将点(x0,y0,z0)的某去心邻域内的点(x,y,z)用向量(x-x0,y-y0,z-z0)的方向余弦及变量t表示为(x0+tcosα,y0+tcosβ,z0+tcosγ),使多元函数f(x,y,z)转化为含自变量t的一元函数f(x0+tcosα,y0+tcosβ,z0+tcosγ),且给出了定理及相应的推论,并给予证明。得出若t→0时f(x0+tcosα,y0+tcosβ,z0+tcosγ)→A是与α,β,γ取值无关的常数,则f(x,y,z)→A((x,y,z)→(x0,y0,z0));若A与α,β,γ取值有关,则(x,y,z)→(x0,y0,z0)时f(x,y,z)的极限不存在。Translating the method of judgement and request the limit of function of many variables into the method of judgment and request the limit of one variable function. Expressing the point (x,y,z) in the vicinity of the point (x_0,y_0,z_0) by direction cosine of the vector (x-x_0,y-y_0,z-z_0) and variable t as (x_0+tcosα,y_0+tcosβ,z_0+tcosγ).Translating function of many variables f(x,y,z)into one、variable function f(x_0+tcosα,y_0+tcosβ,z_0+tcosγ)which only has one independent variable t. And presenting the theorem and deduction correspondingly. Then proves them. If t→0,　f(x_0+tcosα,y_0+tcosβ,z_0+tcosγ)→A, is a constant which is independent of the values of α,β,γ, so f(x,y,z)→A ( (x,y,z)→(x_0,y_0,z_0));if A relate to the values of α,β,γthen the limit of the values of α,β,γis inexistent when(x,y,z)→(x_0,y_0,z_0).

关 键 词：多元函数极限 一元函数极限 与α β γ取值无关 与α β γ取值有关 

分 类 号：O171[理学—数学]