x无穷[(1+x+1/x)^1/2-(1+x)^1/2]x趋于派[(cosx^2+cosx^3)/sinx^3]

Question

x无穷 [(1+x+1/x)^1/2 - (1+x)^1/2]
x趋于派 [(cosx^2 + cosx^3)/sinx^3]

Answer 1

分母有理化
[(1+x+1/x)^1/2 - (1+x)^1/2]
=[(1+x+1/x)^1/2-(1+x)^1/2][(1+x+1/x)^1/2+(1+x)^1/2]/[(1+x+1/x)^1/2+(1+x)^1/2]
=1/{x[(1+x+1/x)^1/2+(1+x)^1/2]}
则lim[(1+x+1/x)^1/2 - (1+x)^1/2]=0
[(cosx^2 + cosx^3)/sinx^3]
=(cosx)^2(1+cosx)/(sinx)^3
=(cosx)^2*2(cosx/2)^2/sinx^3
=2(cosx/2)^2/sinx^3
x趋于派,
可等同于：x趋于0
=2(sinx/2)^2/sinx^3
=2*x^2/4/sinx^3
=0