The Limit Theorem For Asymptotic Analysis Tells Us That If We Have Two Functions

The limit theorem for asymptotic analysis tells us that if we have two functions f(x) and g(x), such that limx!1 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 limx!1 f(x)/g(x) = c, for some finite c? To answer this, it may be useful to look at the definition of big-O.

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