Use L'Hôpital's rule.
If lim_(x->c)f(x)/g(x) is one of the indeterminate forms 0/0 or oo/oo, then lim_(x->c)f(x)/g(x)=lim_(x->c)(f'(x))/(g'(x)), where ' means the derivative with respect to x.
lim_(x->oo)ln^2(x)/x is of the form oo/oo. Thus, lim_(x->oo)(d/dx(ln^2(x)))/(d/dx(x))=lim_(x->oo)(2ln(x))/x.
This is still of the form oo/oo. Apply L'Hôpital's rule again: lim_(x->oo)(2ln(x))/x=lim_(x->oo)(d/dx(2ln(x)))/(d/dx(x))=lim_(x->oo)2/x=0.
This graph demonstrates that ln^2(x)/x does approach 0 as x approaches oo (very slowly).
graph{ln(x)^2/x [-8.89, 8.89, -4.446, 4.446]}
