题目

求lim x趋近于0+ （cos根号x）^(π/x)的极限值..

提问时间：2020-07-25

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).