Question

求极限，要过程

求极限，要过程高等数学

Answer 1

(2)
lim(x->1)[ 1/(1-x) - 3/(1-x^3) ]
=lim(x->1){ 1/(1-x) - 3/[(1-x)(1+x+x^2)] }
=lim(x->1){ [ (1+x+x^2) - 3] /[(1-x)(1+x+x^2)] }
=lim(x->1){ (x+2)(x-1) /[(1-x)(1+x+x^2)] }
=lim(x->1) -(x+2) /(1+x+x^2)
= -(1+2)/(1+1+1)
=-1

(3)
lim(x-> ∞) [(x+2)/(x-1)]^[x^2/(x+1)]
= lim(x-> ∞) [1 + 3/(x-1)]^[x^2/(x+1)]
let
3/(x-1) = 1/y
x= 3y+1

lim(x-> ∞) [(x+2)/(x-1)]^[x^2/(x+1)]
= lim(x-> ∞) [1 + 3/(x-1)]^[x^2/(x+1)]
= lim(y-> ∞) [1 + 1/y]^[(3y+1)^2/(3y+2)]
= lim(y-> ∞) [1 + 1/y]^[(9y^2+3y+9)^2/(3y+2)]
= lim(y-> ∞) [1 + 1/y]^[ 3y -2 + (15/(3y+2))]
= lim(y-> ∞) (1 + 1/y)^(3y)
=e^3