Question

设f(x，y)＝y1+xy?1?ysinπxyarctanx，x＞0，y＞0，求（Ⅰ）g(x)＝limy→+∞f(x，y)；（Ⅱ）limx→0+g(x)

设f(x，y)＝y1+xy?1?ysinπxyarctanx，x＞0，y＞0，求（Ⅰ）g(x)＝limy→+∞f(x，y)；（Ⅱ）limx→0+g(x)．

Answer 1

简单分析一下，答案如图所示

Answer 2

（Ⅰ） g(x)＝

lim

y→+∞

f(x，y)＝

lim

y→∞

(

y

1+xy

?

1?ysin πx y

arctanx

)=

lim

y→∞

(

1

1 y +x

?

1? sin πx y 1 y

arctanx

)＝

1

x

?

1?πx

arctanx

．
（Ⅱ） 

lim

x→0+

g(x)＝

lim

x→0+

(

1

x

?

1?πx

arctanx

)＝

lim

x→0+

arctanx?x+πx2

xarctanx

（通分）=

lim

x→0+

arctanx?x+πx2

x2

＝

lim

x→0+

1 1+x2 ?1+2πx

2x

=

lim

x→0+

?x2+2πx(1+x2)

2x

＝π．