Find the largest interval containing $x = 0$ that the Remainder Estimation Theorem allows over which $f(x)=\sin\!\left(2x\right)$ can be approximated by $p(x)=2x-\frac{\left(2x\right)^{3}}{6}$ to three decimal-place accuracy throughout the interval. Check your answer by graphing $|f(x) - p(x)|$ over the interval you obtained.

Enter Interval in Interval Notation.