Question

高等数学求极限！！第五题

Answer 1

Answer 2

lim(n->∞) n^2. [ 2^(1/n)- 2^(1/(n+1)) ]
consider

lim(x->∞) x^2. [ 2^(1/x)- 2^(1/(x+1)) ]

let
y =1/x

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

y->0
2^y ~ 1 + (ln2) y
2^( y/(y+1) ) ~ 1 + (ln2)( y/(y+1) )

2^y - 2^( y/(y+1) )
~ (ln2) ( y - y/(y+1) )
=(ln2). y^2/(y+1)

lim(x->∞) x^2. [ 2^(1/x)- 2^(1/(x+1)) ]
=lim(y->0) [ 2^y - 2^( y/(y+1)) ] /y^2
=lim(y->0) ln2/(y+1)
=ln2