lim(1/x^2 - cot^2 x ) (x趋近于0) 用洛必达法定则
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lim(1/x^2 - cot^2 x ) (x趋近于0) 用洛必达法定则
![lim(1/x^2 - cot^2 x ) (x趋近于0) 用洛必达法定则](/uploads/image/z/15769336-40-6.jpg?t=lim%EF%BC%881%2Fx%5E2+-+cot%5E2+x+%29+%28x%E8%B6%8B%E8%BF%91%E4%BA%8E0%29+%E7%94%A8%E6%B4%9B%E5%BF%85%E8%BE%BE%E6%B3%95%E5%AE%9A%E5%88%99)
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
=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
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