求极限lim(x+y)sin(1/x²+y²)
4个回答

1°当(x,y)→(0,0)时

|sin[1/(x²+y²)]|≤1

x+y→0

即

sin[1/(x²+y²)]和 x+y分别为(x,y)→(0,0)时的有界量和无穷小量

故

lim[(x,y)→(0,0)]{(x+y)sin[1/(x²+y²)]}=0

2°当(x,y)→(∞,∞)时

|(x+y)sin[1/(x²+y²)]|

≤ (|x|+|y|)[1/(x²+y²)]

=(|x|+|y|)/(x²+y²)

→0

故

lim[(x,y)→(∞,∞)]{(x+y)sin[1/(x²+y²)]}=0

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