Prove that if f, g, and h are functions from R^+ to R^+ such that f(x) = O(g(x))

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In mathematics, big O notation is used to describe the limiting behavior of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions. It is a member of a larger family of notations that is called Landau notation, Bachmann–Landau notation (after Edmund Landau and Paul Bachmann), or asymptotic notation. In computer science, big O notation is used to classify algorithms by how they respond (e.g., in their processing time or working space requirements) to changes in input size. Big O notation characterizes functions according to their growth rates: different functions with the same growth rate may be represented using the same O notation. A description of a function in terms of big O notation usually only provides an upper bound on the growth rate of the function. Associated with big O notation are several related notations, using the symbols o, O, ?, and T, to describe other kinds of bounds on asymptotic growth rates. Example In typical usage, the formal definition of O notation is not used directly; rather, the O notation for a function f(x) is derived by the following simplification rules: If f(x) is a sum of several terms, the one with the largest growth rate is kept, and all others omitted. If f(x) is a product of several factors, any constants (terms in the product that do not depend on x) are omitted. For example, let , and suppose we wish to simplify this function, using O notation, to describe its growth rate as x approaches infinity. This function is the sum of three terms: 6x4, -2x3, and 5. Of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of x, namely 6x4. Now one may apply the second rule: 6x4 is a product of 6 and x4 in which the first factor does not depend on x. Omitting this factor results in the simplified form x4. Thus, we say that f(x) is a big-oh of (x4) or mathematically we can write f(x) = O(x4). One may confirm this calculation using the formal definition: let f(x) = 6x4 - 2x3 + 5 and g(x) = x4. Applying the formal definition from above, the statement that f(x) = O(x4) is equivalent to its expansion, for some suitable choice of x0 and M and for all x > x0. To prove this, let x0 = 1 and M = 13. Then, for all x > x0: so [edit]Usage Big O notation has two main areas of application. In mathematics, it is commonly used to describe how closely a finite series approximates a given function, especially in the case of a truncated Taylor series or asymptotic expansion. In computer science, it is useful in the analysis of algorithms. In both applications, the function g(x) appearing within the O(...) is typically chosen to be as simple as possible, omitting constant factors and lower order terms. There are two formally close, but noticeably different, usages of this notation: infinite asymptotics and infinitesimal asymptotics. This distinction is only in application and not in principle, however—the formal definition for the "big O" is the same for both cases, only with different limits for the function argument.

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