求极限,

题目

求极限,
x趋向于0
求1/x^2-1/(x*tanx)的极限

答案

当x->0时
lim[1/x^2-1/(x*tanx)]
=lim(1/x²-cosx/xsinx)
=lim[1/x²-cosx/(xsinx)]
=lim[(sinx-xcosx)/(x²sinx)]
=lim[(sinx-xcosx)/x³]*lim(x/sinx)
∵lim(x/sinx)=1
又lim[(sinx-xcosx)/x³]
=lim[xsinx/(3x²)] (用一次罗比达法则)
=(1/3)lim(sinx/x)
=1/3
∴lim[1/x^2-1/(x*tanx)]
=(1/3)*1
=1/3.