limx-->0(1/x^2-1/tanx^2)x趋近与0x平方分之一减去tanx平方分之一

Question

limx-->0 (1 /x^2 - 1/tanx^2) x趋近与0 x平方分之一减去tanx平方分之一

Answer 1

lim(x-->0) (1 /x^2 - 1/tanx^2)
=lim(x-->0) (tan^2x-x^2) /(x^2 *tanx^2)
=lim(x-->0) (tan^2x-x^2) /(x^4)
=lim(x-->0) (sin^2x-cos^2x*x^2) /(cos^2x*x^4)
=lim(x-->0) (sin^2x-cos^2x*x^2) /(x^4) (0/0)
=lim(x-->0) (2sinxcosx+2cosxsinx*x^2-2cos^2x*x) /(4x^3)
=lim(x-->0) (sinx+sinx*x^2-cosx*x) /(2x^3) (0/0)
=lim(x-->0) (cosx+cosx*x^2+2xsinx+sinx*x-cosx) /(6x^2)
=lim(x-->0) (cosx*x^2+3xsinx) /(6x^2)
=lim(x-->0) (cosx*x+3sinx) /(6x) (0/0)
=lim(x-->0) (-sinx*x+cosx+3cosx) /(6)
=2/3