Step 1: Identify the tricky parts.
The limit $\lim_{x\to0}\dfrac{|x|\log_e(1+|\sin 2x|)}{x^2(|x|+3)}$ has absolute values and a logarithm, but near $x=0$ we can replace each piece by its leading behaviour.
Step 2: Approximate the sine.
For small $x$, $\sin 2x\approx 2x$, so $|\sin 2x|\approx 2|x|$.
Step 3: Approximate the logarithm.
Since $\log_e(1+t)\approx t$ as $t\to 0$, we get $\log_e(1+|\sin 2x|)\approx 2|x|$.
Step 4: Substitute these into the expression.
The numerator becomes $|x|\cdot 2|x|=2|x|^2=2x^2$, so the fraction is $\dfrac{2x^2}{x^2(|x|+3)}$.
Step 5: Cancel $x^2$.
This leaves $\dfrac{2}{|x|+3}$, which is now perfectly tame at $x=0$.
Step 6: Take the limit.
As $x\to0$, $|x|\to0$, so the value is $\dfrac{2}{0+3}=\dfrac23$. The same value comes from both sides, so the limit exists.
\[ \boxed{\frac23} \]