Real Analysis/Landau notation

../ Landau notation Dedekind's Construction Bibliography __TOC__ The Landau notation is an amazing tool applicable in all of real analysis. The reason it is so convenient and widely used is because it underlines a key principle of real analysis, namely estimation. Loosely speaking, the Landau notation introduces two operators which can be called the "order of magnitude" operators, which essentially compare the magnitude of two given functions. The little-o The little-o provides a function that is of lower order of magnitude than a given function, that is the function is of a lower order than the function . Formally, Definition Let and let Let If then we say that "As , " Examples As , (and ) As , (and ) As , The Big-O The Big-O provides a function that is at most the same order as that of a given function, that is the function is at most the same order as the function . Formally, Definition Let and let Let If there exists such that then we say that "As , " Examples As , As , Applications We will now consider few examples which demonstrate the power of this notation. Differentiability Let and . Then is differentiable at if and only if There exists a such that as , . Mean Value Theorem Let be differentiable on . Then, As , Taylor's Theorem Let be n-times differentiable on . Then, As ,