lim（1/x^2 - cot^2 x ) (x趋近于0) 用洛必达法定则

题目

lim（1/x^2 - cot^2 x ) (x趋近于0) 用洛必达法定则

答案

原式=lim(1/x^2-cos^2x/sin^2x)
=lim(sin^2x-x^2*cos^2x)/(x^2*sin^2x)
=lim(sin^2x-x^2*cos^2x)/(x^4)
=lim[cos^2x(tan^2x-x^2)]/x^4
=lim(tan^2x-x^2)/x^4
=lim(2tanx*sec^2x-2x)/4x^3
=lim(sec^4x+2sec^2x*tan^2x-1)/6x^2
=lim(sec^4x-1)/6x^2+lim(2tan^2x*sec^2x)/6^2x
=lim[(sec^2x+1)(sec^2x-1)]/6x^2+1/3
=lim(sec^2x+1)*tan^2x/6x^2+1/3
=lim(sec^2x+1)/6+1/3
=1/3+1/3
=2/3