求lim(x→0,y→0) ysin(1/xy)的极限
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求lim(x→0,y→0) ysin(1/xy)的极限
![求lim(x→0,y→0) ysin(1/xy)的极限](/uploads/image/z/19691615-47-5.jpg?t=%E6%B1%82lim%EF%BC%88x%E2%86%920%2Cy%E2%86%920%EF%BC%89+ysin%281%2Fxy%29%E7%9A%84%E6%9E%81%E9%99%90)
lim(x→0,y→0) ysin(1/xy)
=(lim(y→0) y)*lim(x→0,y→0) sin(1/xy)
由于lim(y→0) y=0,是无穷小量
|lim(x→0,y→0) sin(1/xy)|≤1,是有界量
根据无穷小量乘以有界量等于无穷小量知
lim(x→0,y→0) ysin(1/xy)=0
=(lim(y→0) y)*lim(x→0,y→0) sin(1/xy)
由于lim(y→0) y=0,是无穷小量
|lim(x→0,y→0) sin(1/xy)|≤1,是有界量
根据无穷小量乘以有界量等于无穷小量知
lim(x→0,y→0) ysin(1/xy)=0
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