Question

[1/x-arctan(1/x)] / (sin1/x)^3 ,x趋于无穷,请问怎么解?

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