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let
y=1/x
y->0
(1-y^3)^(1/3) ~ 1 - (1/3)y^3
1-(1-y^3)^(1/3) ~ (1/3)y^3
lim(x->∞) [ x + (1-x^3)^(1/3) ]
=lim(y->0) [ 1/y + (1- 1/y^3)^(1/3) ]
=lim(y->0) [ 1 + ( y^3 -1)^(1/3) ]/y
=lim(y->0) [ 1 - ( 1-y^3 )^(1/3) ]/y
=lim(y->0) (1/3)y^3/y
=0
y=1/x
y->0
(1-y^3)^(1/3) ~ 1 - (1/3)y^3
1-(1-y^3)^(1/3) ~ (1/3)y^3
lim(x->∞) [ x + (1-x^3)^(1/3) ]
=lim(y->0) [ 1/y + (1- 1/y^3)^(1/3) ]
=lim(y->0) [ 1 + ( y^3 -1)^(1/3) ]/y
=lim(y->0) [ 1 - ( 1-y^3 )^(1/3) ]/y
=lim(y->0) (1/3)y^3/y
=0
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