Big O Notation question

I am trying to understand the Big-O and little-O notation, so I plotted 2 graphs which I have posted below, but I still dont really get the concept of it. What exactly does the $O\left(\frac{1}{x^6}\right)$ term do?

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Here is an excellent explanation of Big O from Stackoverflow: stackoverflow.com/questions/487258/… –  Matt May 5 '12 at 23:42

The big-O term in your example stands for a function whose absolute value is less than a constant times $x^{-6}$. So, when $x$ is large, the term is essentially negligible since $x^{-6}$ is small. That's why the green graph and the red graph seem to converge as $x$ increases. There is a function there which is hidden by the big-O; you don't know exactly what it is, only what it's bounded by.

More specifically, if you write, for example,

$$f(x) = \frac{x^{-1} + O\left(x^{-2}\right)}{e^x} \quad (x \to \infty)$$

you're saying that there is a function $g$ and constants $C$ and $x_0$ satisfying $|g(x)| \leq C \left|x^{-2}\right|$ when $x \geq x_0$ such that

$$f(x) = \frac{x^{-1} + g(x)}{e^x}.$$

The formula for $g(x)$ may be massive, unwieldy, or otherwise distracting. If all you care about is what it's bounded by, then you can write the big-O symbol in its place.

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Thanks Antonio, appreciated! –  Jonathan May 12 '12 at 16:16