I was watching a lecture on Asymptotic Expansions, and there was a part where:
$\arctan(\epsilon)$ was said to be approximately $\epsilon -\frac{\epsilon^3}{3}+\frac{\epsilon^5}{5}+...$ (Taylor series centered at 0). With the approximation being more accurate with how small $\epsilon$ is.
We can then take $\arctan(\epsilon) = \epsilon + R$ where $R$ is the error (i.e. remaining terms of the infinite series).
Then the presenter says that for this approximation to be accurate, there's a sense in which $R$ must be negligible compared to the approximation term $\epsilon$. He writes $R \ll \epsilon$.
Alternatively, he says $\frac{R}{\epsilon} \to 0$ as $\epsilon \to 0$. Can you help me understand why these two statements are equivalent? $R \ll \epsilon$ and $\frac{R}{\epsilon} \to 0$ as $\epsilon \to 0$.
Thanks!