求limx→∞，y→a(cosyx)x3x+y3．

Question

求

lim

x→∞，

y→a

(cos(y)/(x))(x3)/(x+y3)．

Answer 1

由于

lim

x→∞，y→a

(cos(y)/(x))(x3)/(x+y3)=

lim

x→∞，y→a

[(1+cos(y)/(x)-1)(1)/(cos(y)/(x)-1)](cos(y)/(x)-1)(x3)/(x+y3)=e

lim

x→∞，y→a

(cos(y)/(x)-1)(x3)/(x+y3)，
而

lim

x→∞，y→a

(cos(y)/(x)-1)(x3)/(x+y3)=

lim

x→∞，y→a

(-2sin2(y)/(x))(x3)/(x+y3)
=-2

lim

x→∞，y→a

(y2)/(x2)•(x3)/(x+y3)=-2

lim

x→∞，y→a

(y2)/(1+(y3)/(x))=-2a2，
故

lim

x→∞，y→a

(cos(y)/(x))(x3)/(x+y3)=e-2a2．