The limit theorem for asymptotic analysis tells us that if we have two functions f(x) and g(x), such that lim x-inf f(x)/g(x) = c, for some finite

The limit theorem for asymptotic analysis tells us that if we have two functions f(x) and g(x), such that lim x->inf f(x)/g(x) = c, for some finite c, then f(x) = O(g(x)). Is the opposite true? That is, if f(x) = O(g(x)), must it be the case that lim x->inf f(x)/g(x) = c, for some finite c? To answer this, it may be useful to look at the definition of big-O. Explain your answer.