Question

下面这个二元函数极限存在吗？如果存在怎么证明？

是重极限

Answer 1

存在,分三种情况证明
当x=y时，显然lim (x+y)/根号(x^3 +y^3) = lim 2x/根号(2x^3) =0
当x<y时，lim (x+y)/根号(x^3 +y^3) = lim [(x/y) /y^(1/2) + 1/y^(1/2)]/根号(1+(x/y)^(3/2) >= lim [(x/y) /y^(1/2) + 1/y^(1/2)]/根号(2) =0

lim [(x/y) /y^(1/2) + 1/y^(1/2)]/根号(1+(x/y)^(3/2) <= lim [(x/y) /y^(1/2) + 1/y^(1/2)]/根号(1) =0

当x>y时证明和第二种一样

Answer 2

令y=kx，
lim=lim(1+k)/√(1+k³)x
因为x.y都趋于+∞，所以k≠－1，所以lim=0，存在

Answer 3

A = (x+y)/(x^3+y^3)^0.5

A^2 = (x+y)^2/(x^3+y^3)
=（x+y）^2/[(x+y)(x^2-xy+y^2)]
=(x+y)/((x-y)^2+xy)
<=(x+y)/xy=1/x+1/y
0=<A<=(1/x+1/y)^0.5
夹逼 极限=0