Question

求这个极限题。

Answer 1

let
y=π-x
x->π , y->0
y->0
分母
e^[(siny)^2] -1 ~ e^(y^2) -1 ~ y^2
分子
((π-y)^3-π^3) = -y [ (π-y)^2 +π(π-y) + π^2 ]
[(π-y)^3-π^3]. sin5y ~ -5y^2 . [ (π-y)^2 +π(π-y) + π^2 ]
--------------
lim(x->π) (x^3-π^3).sin5x/{ e^[(sinx)^2] -1 }
=lim(y->0) [(π-y)^3-π^3]. (sin5y)/{ e^[(siny)^2] -1 }
=lim(y->0) [(π-y)^3-π^3]. sin5y/{ e^[(siny)^2] -1 }
=lim(y->0) -5y^2 . [ (π-y)^2 +π(π-y) + π^3 ]/ y^2
=lim(y->0) -5[ (π-y)^2 +π(π-y) + π^2]
=-5[ π^2 +π^2 + π^2 ]
=-15π^2

Answer 2