lim(n->∞) [ 2^(1/n) - 2^(1/(n+1) ) ]
= ( 2^0 -2^0 )
=0
lim(x->∞) [ sin(2/x)+cos(1/x) ]^x
let
y=1/x
lim(x->∞) [ sin(2/x)+cos(1/x) ]^x
=lim(y->0) [ sin(2y)+cosy ]^(1/y)
x->0
sin2x +cosy ~ 1+2y
lim(x->∞) [ sin(2/x)+cos(1/x) ]^x
=lim(y->0) [ sin(2y)+cosy ]^(1/y)
=lim(y->0) ( 1+ 2y )^(1/y)
=lim(z->0) ( 1+ z )^(2/z) ( z=2y)
=e^2
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