求lim x趋近于0+ (cos根号x)^(π/x)的极限值..

求lim x趋近于0+ (cos根号x)^(π/x)的极限值..

题目
求lim x趋近于0+ (cos根号x)^(π/x)的极限值..
答案
附注:
(1)下面用到的知识至少至洛必达法则;
(2)下面用等号连接式子并非完全合适,至少更为合适的方式是用推导符号连接.
lim(x->0+)((cos(x^(1/2)))^(pi/x))
=lim(x->0+)e^(ln((cos(x^(1/2)))^(pi/x)))
=e^(lim(x->0+)ln((cos(x^(1/2)))^(pi/x)))
=e^(lim(x->0+)(pi/x)ln(cos(x^(1/2))))
=e^(lim(x->0+)(pi/x)(1-cos(x^(1/2)))(1/(1-cos(x^(1/2))))ln(1-(1-cos(x^(1/2)))))
=e^(lim(x->0+)(pi/x)(1-cos(x^(1/2)))ln(((1-(1-cos(x^(1/2)))))^(1/(1-cos(x^(1/2))))))
=e^((lim(x->0+)pi)*(lim(x->0+)(1-cos(x^(1/2)))/x)*(lim(x->0+)ln(((1-(1-cos(x^(1/2)))))^(1/(1-cos(x^(1/2)))))))
=e^(pi*(lim(x->0+)(1-cos(x^(1/2)))'/(x)')*(lim((1-cos(x^(1/2)))->0+)ln(((1-(1-cos(x^(1/2)))))^(1/(1-cos(x^(1/2)))))))
=e^(pi*(lim(x->0+)(sin(x^(1/2))*(1/2)/(x^(1/2)))/1)*lne)
=e^(pi*((1/2)*(lim(x->0+)sin(x^(1/2))/(x^(1/2))))*lne)
=e^((pi/2)*(lim((x^(1/2))->0+)sin(x^(1/2))/(x^(1/2))))
=e^((pi/2)*1)
=e^(pi/2)
答:式子=e^(pi/2).
举一反三
我想写一篇关于奥巴马的演讲的文章,写哪一篇好呢?为什么好
奥巴马演讲不用看稿子.为什么中国领导演讲要看?
想找英语初三上学期的首字母填空练习……
英语翻译
1,人们染上烟瘾,最终因吸烟使自己丧命.
最新试题
热门考点

超级试练试题库

© 2017-2019 超级试练试题库,All Rights Reserved.