例2.16分子的积分怎么算啊

例2.16分子的积分怎么算啊

这种含变限积分的极限一般不用算出积分，直接使用洛必达就可以解决，具体如下



追问积分是怎么使用洛必达的，哪一块的知识点啊？追答变限积分求导，牛顿莱布尼兹公式追问想起来啦！谢谢！

别求积分，用洛必达法则。分子求导是用变限积分的求导法则

lim(x->+∞) ∫(1->x) [ t^2.(e^(1/t) -1 ) -t ] dt /[ x^2.ln(1+ 1/x) ] (∞/∞)
=lim(x->+∞) [ x^2.(e^(1/x) -1 ) -x ] /[ x^2. ( 1/(x+1) -1/x) + 2xln(1+ 1/x) ]
=lim(x->+∞) [ x(e^(1/x) -1 ) -1 ] /[ x ( 1/(x+1) -1/x) + 2ln(1+ 1/x) ]
=lim(x->+∞) [ x(e^(1/x) -1 ) -1 ] /[ -1/(x+1) + 2ln(1+ 1/x) ]

let
y= 1/x
lim(x->+∞) [ x(e^(1/x) -1 ) -1 ] /[ -1/(x+1) + 2ln(1+ 1/x) ]
=lim(y->0) [ (1/y)(e^y -1 ) -1 ] /[ -y/(y+1) + 2ln(1+ y) ]
=lim(y->0) [ (1/y)(e^y -1 ) -1 ] /[ -1 + 1/(y+1) + 2ln(1+ y) ]
y->0

分子
e^y ~ 1+y +(1/2)y^2
(1/y)(e^y -1 ) ~ 1 + (1/2)y
(1/y)(e^y -1 ) -1 ~ (1/2)y
分母
1/(y+1) ~ 1-y
-1+1/(y+1) ~ -y
2ln(1+y) ~ 2y

-1+1/(y+1) +2ln(1+y) ~ y
lim(x->+∞) ∫(1->x) [ t^2.(e^(1/t) -1 ) -t ] dt /[ x^2.ln(1+ 1/x) ]
=lim(y->0) [ (1/y)(e^y -1 ) -1 ] /[ -1 + 1/(y+1) + 2ln(1+ y) ]
=lim(y->0) (1/2)y/y
=1/2