Alternative geometric interpretation for big-o and little-o

Question

I understand that, in big-o notation, when we say that a function $f$ is $O(x^2)$ we're basically saying that

$$|f(x)|\le M |x^2|$$ for some constant $M>0$ and for all $x>x_0$ for some $x_0$

Now, for little-o, if we say that $g(x)$ is $o(x^2)$, we're saying that:

$$|g(x)|\le \epsilon|x^2|$$

for all constant $\epsilon>0$ and for all $x>x_0$ for some $x_0$

Now, in this book, it visualizes $o(f(x)$ and $O(f(x)$ radically different:

I can't understand what exactly is a graph of an $O(h)$ mapping. Is it a class of graphs of functions, or whatever? Also, it says it must cross the origin in all 3 visualizations. Could somebody give me a better explanation, or a visualization with real functions?

Answer 1

But $O$ and $o$ are under-specified without a statement such as $x\to\infty$ or $x\to 0$.

For example: "$x^2=O(x)$ as $x\to 0$" is true; but "$x^2=O(x)$ as $x\to \infty$" is false.

The intepretation you quoted at the beginning is about $x\to \infty$. In some areas, like computer science, people tend to consider this situation exclusively, so "$x\to\infty$", is understood without saying.

But the book is talking about $x\to 0$, which is a different situation; we are considering $x$ near $0$, not large $x$.

• $f(x) = O(x)$ as $x\to 0$ means: there is $C$ such that $|f(x)|\le C|x|$ for all $x$ sufficiently close to $0$. Such a function might look like this:

That is, its graph is contained in some bowtie-shaped region like the one between the red lines.

• $f(x) = o(x)$ as $x\to 0$ means: for any $c>0$ the inequality $|f(x)|\le c|x|$ holds for all $x$ sufficiently close to $0$. Such a function might look like this:

where I have shown that in smaller neighborhoods of $0$, one can use flatter bowtie (green).