# Real Analysis/Landau notation

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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*[edit]

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[edit]

Let and let

Let

If then we say that

"As , "

### Examples[edit]

- As , (and )
- As , (and )
- As ,

## The Big-*O*[edit]

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[edit]

Let and let

Let

If there exists such that then we say that

"As , "

### Examples[edit]

- As ,
- As ,

## Applications[edit]

We will now consider few examples which demonstrate the power of this notation.

### Differentiability[edit]

Let and .

Then is differentiable at if and only if

There exists a such that as , .

### Mean Value Theorem[edit]

Let be differentiable on . Then,

As ,

### Taylor's Theorem[edit]

Let be *n*-times differentiable on . Then,

As ,