当x趋向于0时,求极限(1/x)^2-(cotx)^2

题目

当x趋向于0时,求极限(1/x)^2-(cotx)^2

答案

=lim (1/x)^2-(1/tan x)^2
=lim (x^2 - tan^2 x) /(x^2 · tan^2 x)
=lim (x^2 - tan^2 x) /(x^4) 【等价无穷小代换】
=lim (2x - 2 tan x /cos²x) /(4x^3) 【洛比达法则】
=lim (1/cos^3 x) ·lim (x·cos^3 x - sin x) /(2x^3)
=1·lim (cos^3 x - 3 x·cos^3 x·sin x - cos x) /(6x^2) 【洛比达法则】
=lim (cos x) ·lim [(cos^2 x - 1 ) - 3 x·cos^2 x·sin x] /(6x^2)
=1·lim [ -sin^2 x - 3 x·cos^2 x·sin x] /(6x^2)
= -lim sinx·[ sin x + 3 x·cos^2 x] /(6x^2)
= -lim x·[ sin x + 3 x·cos^2 x] /(6x^2) 【等价无穷小代换】
= -lim [ sin x + 3 x·cos^2 x] /(6x)
= -lim [ cos x + 3 (cos^2 x -2·x·cosx·sinx) ] /6【洛比达法则】
= -[ 1 + 3 (1 -2×0) ] /6
= -2/3