多元函数极限中几个问题的释疑

计算多元函数的极限时,许多情况下可以应用等价无穷小、两边夹法则等方法。如果多元函数的极限不存在,经常讨论动点以不同路径趋于定点,而函数以不同的趋势变化,得出极限不存在的结论。经常选取的路径有y=kx,或者计算两个不相等的二次极限等。在计算多元函数的极限时,由于动点的变化方向、方式复杂多样,选取不同的路径用来分析函数的不同变化趋势,或者计算两个不相等的二次极限,能否得出多元函数极限不存在的结论,与聚点邻域的形状有关。本文对计算多元函数极限的几个问题作了初步的探讨。

Discussion of Several Issues about Limit of Multivariate Function

When calculating the limit of multivariate function, the equivalent infinitesimal and sandwich theorem can be used in many cases. If the limit of multivariate function does not exist, we often discussed fixed point to a different path which tends to be a fixed point, the function with different trends changing, it can conclude that limit does not exist, the frequently selected path is y=kx, or calculating the two is not equal to the secondary limit. When calculating the limit of multivariate function, due to the complex method, select a different path for trend analysis function or calculation of the two different secondary limit, can be drawn the conclusion that limit does not exist, related with the shape of the poly-point neighborhood. The paper discuss some problem of limit of multivariate function.