In the taylor series for sin(x), we write:

$$ \sin{x} = x - \frac{x^3}{6} + \frac{x^5}{120} + O(x^7) $$

Meaning that $\sin{x} = x - \frac{x^3}{6} + \frac{x^5}{120}$ and terms of order $x^7$ and higher, so we say that those 'higher order terms' are equal to $O(x^7)$.

However, according to wikipedia, the definition of $f(x) = O(g(x))$ is that for all $x > x_o$ for some $x_o$, $\frac{|f(x)|}{|g(x)|} < M $ for some constant M. According to this definition, the terms after the $x^7$th term in the taylor expansion of $\sin{x}$ are /not/ $O(x^7)$, because as $x$ approaches infinity, the higher order terms should dominate the $O(x^7)$ term, not be bounded by it.

Am I missing something here?