Question

求极限limx→0 1-cosx²/x的三次方sinx

Answer 1

当x→0时，1-cosx^2~x^4/2,sinx~x, 等价无穷小代换
lim【x→0】( 1-cosx²)/(x的三次方sinx)=lim【x→0】（x^4/2）/(x^4)=1/2

也可以用洛必达法则解
lim【x→0】( 1-cosx²)/(x的三次方sinx)

=limx→0 2xsinx²/(3x^2sinx+x^3cosx)
=limx→0 2sinx²/(3xsinx+x^2)
=limx→0 4xcosx²/(3sinx+3xcosx+2x)
=limx→0 4cosx²/(3sinx/x+3cosx+2)
=4/8
=1/2

Answer 2

利用无穷小等量代换可知
1-cosx²~1/2x⁴
sinx~x
带入可知
=lim1/2x⁴ /(x³*x)
=1/2