y=1/x
lim(x->+∞腔册) ( x^3+5x^2+4x+3)^(1/3) - (x^3+x^2+3x+2)/[x.√(x^2+x+1)]
=lim(y->0+) ( 1/y^3+5/y^2+4/y+3)^(1/3)
胡烂 - (1/y^3+1/y^2+3/y+2)/[ (1/y).√(1/y^2+1/y+1)]
=lim(y->0+) ( 1+5y+4y^2+3y^3)^(1/3) /y
- (1+y+3y^2+2y^3)/[ y.√(1+y+y^2)]
通分伍做宏母
=lim(y->0+) [( 1+5y+4y^2+3y^3)^(1/3).√(1+y+y^2)- (1+y+3y^2+2y^3) ]
/[ y.√(1+y+y^2)]
lim(y->0+) √(1+y+y^2) =1
=lim(y->0+) [( 1+5y+4y^2+3y^3)^(1/3).√(1+y+y^2)- (1+y+3y^2+2y^3) ] /y
=lim(y->0+) (7/6)y /y
=7/6
//
等价交换
y->0+
分子
( 1+5y+4y^2+3y^3)^(1/3).√(1+y+y^2)
=[1 +(5/3)y+o(y) ] . [ 1+(1/2)y +o(y) ]
=1 + (13/6)y + o(y)
( 1+5y+4y^2+3y^3)^(1/3).√(1+y+y^2)- (1+y+3y^2+2y^3) = (7/6)y + o(y)
2024-10-13 广告