Question

求下列极限，打勾的，俩题。谢谢(*°∀°)=3

Answer 1

(1)
lim(x->1) (x-x^x)/(1-x+lnx) (0/0)
=lim(x->1) (1-(1+lnx)x^x)/(-1+ 1/x)
=lim(x->1) x(1-(1+lnx)x^x)/(-x+ 1)
=lim(x->1) -x(1-(1+lnx)x^x)/(x- 1) (0/0)
=lim(x->1) { -(1-(1+lnx)x^x) + [x(1+lnx)^2 +1) . x^x }
= 2

(2)
lim(x-> ∞) (sin(1/x) +cos(1/x) )^x
let
y=1/x

lim(x-> ∞) (sin(1/x) +cos(1/x) )^x
=lim(y-> 0) (siny +cosy )^(1/y)

y->0
siny +cosy ~ 1 +y

lim(x-> ∞) (sin(1/x) +cos(1/x) )^x
=lim(y-> 0) (siny +cosy )^(1/y)
=lim(y-> 0) (1+y )^(1/y)
=e

追问

谢谢