Question

1/x(cotx-1/x)x趋近0时极限

Answer 1

简单计算一下即可，答案如图所示

Answer 2

lim(x趋于0)(1/x) · (cotx-1/x)
=lim(x趋于0)(1/x) · [(x-tanx)/xtanx]
=lim(x趋于0)(1/x) · [(x-tanx)/x^2] 等价无穷小
=lim(x趋于0)[(x-x^3/3+o)/x^3] 泰勒展开
= -1/3

Answer 3

原式=limx→0 (x-tanx)/(x^2*tanx)
=limx→0 (x-tanx)/x^3(tanx~x,等价无穷小替换)
=limx→0 -tan^2x/2x^2(罗比塔法则)
=limx→0 -x^2/2x^2(tanx~x,等价无穷小替换)
=-1/2.