求lim x趋近于0+ (cos根号x)^(π/x)的极限值..
来源:学生作业帮 编辑:作业帮 分类:数学作业 时间:2024/08/23 18:37:17
求lim x趋近于0+ (cos根号x)^(π/x)的极限值..
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).
(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).
求lim x趋近于0+ (cos根号x)^(π/x)的极限值..
当x趋近于0,求lim(1-sinx)^1/x的极限值.
lim[cos(a/x)]^(x^2) x趋近于无穷大时求极限值
lim e 指数是x分之1 当x趋近于0+的时候求极限值
x趋近于+0求极限lim(cos√x)^π/x
利用简单方法求极限.(1)x趋近于0,lim[(1/sin平方x)-1/x平方] (2)x趋近于0+,(cos根号X)的
lim cos x/x 的极限值[x趋向于无穷大]
lim x趋近于0 (sinx+tan3x)/2x的极限值
求limx趋近于0 1-根号cosx/x^2 的极限值
求lim(x趋近于0)sin根号x
x趋近于0,lim(1-根号下cosx)/x(1-cos根号下x)
求lim(x趋近于1)(x的根号m次方-1)/(x-1)