[1/x-arctan(1/x)]/(sin1/x)^3,x趋于无穷,请问怎么解?RT原题:limx→无穷[1/x-arctan(1/x)]/[sin(1/x)]^3

Question

[1/x-arctan(1/x)] / (sin1/x)^3 ,x趋于无穷,请问怎么解?
RT
原题:
lim x→无穷 [1/x-arctan(1/x)] / [sin(1/x)]^3

Answer 1

令t=1/x
则t->0
lim x→无穷 [1/x-arctan(1/x)] / [sin(1/x)]^3
=lim t→0 [t-arctan(t)] / [sin(t)]^3
等效替换sin t
=lim t→0 [t-arctan(t)] /t^3
0/0
洛必达
=lim t→0 [1-1/(1+t^2)] /3t^2
=lim t→0 [t^2/(1+t^2)] /3t^2
=lim t→0 1/[3(1+t^2)]
=1/(3*1)
=1/3