作业帮多元函数的极限 [lim xln(1+xy)]/(x+y) x→0,y→0 极限是否存在?
来源：学生作业帮助网 编辑：作业帮 时间：2024/04/28 07:14:33
![多元函数的极限 [lim xln(1+xy)]/(x+y) x→0,y→0 极限是否存在?](/uploads/image/z/7649428-4-8.jpg?t=%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0%E7%9A%84%E6%9E%81%E9%99%90+%5Blim+xln%281%2Bxy%29%5D%2F%28x%2By%29+x%E2%86%920%2Cy%E2%86%920+%E6%9E%81%E9%99%90%E6%98%AF%E5%90%A6%E5%AD%98%E5%9C%A8%3F)
多元函数的极限 [lim xln(1+xy)]/(x+y) x→0,y→0 极限是否存在?
多元函数的极限 [lim xln(1+xy)]/(x+y) x→0,y→0 极限是否存在?
多元函数的极限 [lim xln(1+xy)]/(x+y) x→0,y→0 极限是否存在?
lim(x->0) xln(1+xy)/(x+y)=0
lim(y->0) xln(1+xy)/(x+y)=0
lim(x->0+,y->0)=lim(x->0-,y->0)=0
极限存在
不存在。令y=-x x^3
x→0，y→0，A = [lim xln(1+xy)]/(x+y) = [lim (x*xy)]/(x+y) = lim[(x^2*y)/(x+y)]
f(x,y) = (x^2*y)/(x+y)
x→0，y=kx ,limf(x,y) =lim(k*x^3)/(x+kx) = lim[(k/(1+k)]*x^2 = 0
由此假设：x→0，y→0，limf(x,y) ...
全部展开
x→0，y→0，A = [lim xln(1+xy)]/(x+y) = [lim (x*xy)]/(x+y) = lim[(x^2*y)/(x+y)]
f(x,y) = (x^2*y)/(x+y)
x→0，y=kx ,limf(x,y) =lim(k*x^3)/(x+kx) = lim[(k/(1+k)]*x^2 = 0
由此假设：x→0，y→0，limf(x,y) = 0
试用定义证明
|f(x,y) - 0| = | (x^2*y)/(x+y) - 0 |
=x^2* |y| / |x+y|
<= (x^2+y^2) *|y| /|x+y|
<= x^2+y^2
可见任意a>0，取b=(a)^(1/2) ，则当
0<[(x-0)^2 + (y-0)^2] ^(1/2) < b
即P(x,y)属于f(x,y)定义域内（0,0）的去心邻域时，总有 |f(x,y) - 0| < a
成立，所以 x→0，y→0 ,limf(x,y) = 0
即x→0，y→0，[lim xln(1+xy)]/(x+y) = 0
收起