Real Analysis/Landau notation

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 $$o(g(x))$$ is of a lower order than the function $$g(x)$$. Formally,

Definition
Let $$A\subseteq\mathbb{R}$$ and let $$c\in\mathbb{R}$$

Let $$f,g:A\to\mathbb{R}$$

If $$\lim_{x\to c}\frac{f(x)}{g(x)}=0$$ then we say that

"As $$x\to c$$, $$f(x)=o(g(x))$$"

Examples

 * As $$x\to\infty$$, (and $$m0$$ such that $$\lim_{x\to c}\left| \frac{f(x)}{g(x)}\right| <M$$ then we say that

"As $$x\to c$$, $$f(x)=O(g(x))$$"

Examples

 * As $$x\to 0$$, $$\sin x=O(x)$$
 * As $$x\to \tfrac{\pi}{2}$$, $$\sin x=O(1)$$

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

Differentiability
Let $$f: \mathcal{U} \subseteq \mathbb{R} \to\mathbb{R}$$ and $$ x_0 \in \mathcal{U}$$.

Then $$f$$ is differentiable at $$x_0$$ if and only if

There exists a $$\lambda \in\mathbb{R}$$ such that as $$x\to x_0$$, $$f(x) = f(x_0) + \lambda(x-x_0)+o\left( |x-x_0|\right)$$.

Mean Value Theorem
Let $$f:[a,x]\to\mathbb{R}$$ be differentiable on $$[a,b]$$. Then,

As $$x\to a$$, $$f(x)=f(a)+O(x-a)$$

Taylor's Theorem
Let $$f:[a,x]\to\mathbb{R}$$ be n-times differentiable on $$[a,b]$$. Then,

As $$x\to a$$, $$f(x)=f(a)+\tfrac{(x-a)f'(a)}{1!}+\tfrac{(x-a)^2f''(a)}{2!}+\ldots+\tfrac{(x-a)^{n-1}f^{(n-1)}(a)}{(n-1)!}+O\left( (x-a)^n\right)$$