Question

高数，这个怎么做？

划线的部分

Answer 1

lim(x->0) ∫(0->sinx) (1+t)^(1/t) dt / ∫(0->5x) (sint/t ) dt
(0/0 分子分母分别求导)
=lim(x->0) (cosx) (1+sinx)^(1/sinx) / [5 (sin5x)/(5x) ]
=(1/5) lim(x->0) (1+sinx)^(1/sinx)
=(1/5) lim(x->0) (1+x)^(1/x)
=e/5

追问

倒数第三行怎样到第二行的？

人哪去了？

喂喂喂

追答

g(x)= ∫(0->sinx) (1+t)^(1/t) dt
g'(x) = (1+sinx)^(1/sinx) . d/dx sinx = cosx. (1+sinx)^(1/sinx)
h(x) =∫(0->5x) (sint/t ) dt
h'(x) = ( sin(5x)/(5x) ) . d/dx (5x) = 5( sin(5x)/(5x) )
lim(x->0) ∫(0->sinx) (1+t)^(1/t) dt / ∫(0->5x) (sint/t ) dt
(0/0 分子分母分别求导)
=lim(x->0) (cosx) (1+sinx)^(1/sinx) / [5 (sin5x)/(5x) ]
=lim(x->0) (1+sinx)^(1/sinx) .lim(x->0) (cosx) . lim(x->0) {1/ [5 (sin5x)/(5x) ] }
=[lim(x->0) (1+sinx)^(1/sinx) ]. (1) . (1/5)
=(1/5) lim(x->0) (1+sinx)^(1/sinx)