洛必达法则完全证明.docx

《洛必达法则完全证明.docx》由会员上传分享，免费在线阅读，更多相关内容在教育资源-天天文库。

1、⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯料推荐⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯洛必达法则完全证明定理1limf(x)limg(x)0，limf'(x)存在或为xx0xx0xx0g'(x)，则limf(x)=limf'(x)xx0g(x)xx0g'(x)证明见经典教材。定理2limf(x)limg(x)xxt1x1证明：limf(x)limf()xt0t0，limf'(x)存在或为xx0g'(x)1t10，limxg(x)limg()xt0tf(x)f'(x)，则lim=lim0，由定理1txf(1)f'(1)(12)f'(1)xt11limf(x)=limtlimttlimt

2、limf'(x)。xg(x)t01t011t01xg'(x)g(t)g'(t)(t2)g'(t)定理3limf(x)limg(x)，limf'(x)存在或为，则limf(x)=limf'(x)xx0xx0xxg'(x)xxg(x)xx0g'(x)001证明：limf(x)=limg(x)，由定理1xx0g(x)xx01f(x)1limf(x)=limg(x)=limxx0g(x)xx01xx0f(x)g'(x)g2(x)f'(x)f2(x)lim((f(x))2g'(x))xx0g(x)f'(x)1）设limf(x)存在且不为0，则xx0g(x)limf(x)lim(f(x))2limg'

3、(x)，limf(x)limf'(x)xx0g(x)xx0g(x)xx0f'(x)xx0g(x)xx0g'(x)2）设limf(x)存在且为0，设k0，则xx0g(x)lim(f(x)k)0xx0g(x)有lim(f(x)k)=limf(x)+kg(x)xx0g(x)xx0g(x)1⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯料推荐⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯⋯f(x)，g(x)是不同阶无穷大，f(x)+kg(x)仍为无穷大，由1）lim(f(x)k)=limf(x)+kg(x)=limf'(x)+kg'(x)=lim(f'(x)+k)xx0g(x)xx0g(x)xx0g'(x)xx

4、0g'(x)limf(x)=limf'(x)xx0g(x)xx0g'(x)3）设limf(x)=，则limg(x)=0，由2）得xx0g(x)xx0f(x)limg(x)=limg'(x)=0，limf(x)=limf'(x)=xx0f(x)xx0f'(x)xx0g(x)xx0g'(x)综合1）2）3）定理3证毕。定理4limf()limg(x)，limf'(x)存在或为，则limf(x)f'(x)xxx=limxx0g'(x)xx0g(x)xx0g'(x)证明方法类似定理2。2