123222)3arcsin(),(yxyxyxf 013222yxyx 22242yxyx所求定义域为所求定义域为.,42|),(222yxyxyxD 4yxyxz224224yx 422 yxyxyx 0y);,(yxD422yxyx0,yxyo56Ayxfyyxx),(lim00Ayxfyxyx),(lim),(),(00或APfPP)(lim0或APfPP)(lim07801sin)(lim2222)0,0(),(yxyxyx01sin)(2222 yxyx22221sinyxyx 22yx ,0 ,当当 时，时，22)0()0(0yx 01sin)(2222yxyx结论成立结论成立9xyyxxyxyxexyxyxyyx3sin)(lim)3)sin(lim)2lim)122)0,0(),()5,0(),(220110.)sin(lim222)0,0(),(yxyxyx 222)0,0(),()sin(limyxyxyx,)sin(lim22222)0,0(),(yxyxyxyxyx 其中其中yxyxyx22)0,0(),()sin(limuuusinlim0,1 222yxyx x21,00 x.0)sin(lim22200 yxyxyxyxu2 11时，该函数极限不存在当证明：二元函数)0,0(),()0,0(),(0)0,0(),()(),(22yxyxyxyxxyyxf12263)0,0(),(limyxyxyx 取取,3kxy 263)0,0(),(limyxyxyx 6263303limxkxkxxkxyx ,12kk 其值随其值随k的不同而变化，的不同而变化，故极限不存在故极限不存在13)()(lim00PfPfPP14 0,00,),(222222yxyxyxxyyxf22)0,0(),(limyxxyyx 22220limxkxkxkxyx 21kk 15 )0,0(),(,0)0,0(),(,),(2233yxyxyxyxyxf),(lim)0,0(),(yxfyxyyxyxyxxyxyx222)0,0(),(222)0,0(),(limlim0lim,0lim222)0,0(),(222)0,0(),(yyxyxyxxyxyx)0,0(0),(lim)0,0(),(fyxfyx16.11lim)0,0(),(xyxyyx)11(11lim)0,0(),(xyxyxyyx原原式式111lim00 xyyx.21 。