Question

求大神帮忙解一下这5道题！万分感谢！

Answer 1

(1)
lim(n->∞) ( 1+ 1/2+...+1/2^n) /( 1+ 1/3+...+1/3^n)
=lim(n->∞) 2[ 1- 1/2^(n+1) ] / { (3/2)[ 1- 1/3^(n+1)] }
= 2/ ( 3/2)
=4/3
(2)
lim(n->∞) ( e^n -1) /( e^(2n) +1)
分子分母同时除以 e^(2n)
lim(n->∞) [ 1/e^n -1/e^(2n) ] /[ 1+ 1/e^(2n) ]
=0
(3)
lim(n->∞) [ 2^(n+1) +3^(n+1) ] /( 2^n +3^n)
分子分母同时除以 3^n
= lim(n->∞) [ 2.(2/3)^n +3 ] /[ (2/3)^n +1 ]
=(0+3)(0+1)
=3
(4)
|sin(1/x)|≤1 and lim(x->0) x^2 =0
=> lim(x->0) x^2.sin(1/x) =0
(5)
lim(x->-∞) e^x/ arctanx
= 0/(-π/2)
=0