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To compute the limit as n goes to infinity of f(n)/g(n), where f(n) and g(n) either both go to zero or both grow without limit, compute the limit as n goes to infinity of f'(n)/g'(n), where f'(n) and g'(n) are the derivatives of f(n) and g(n) with respect to n.
See http://www.karlscalculus.org/calc8_0.html for many worked examples.