The value of the limit:

$\lim_{x\to 0}\frac{|x|\log_e(1+|\sin 2x|)}{x^2(|x|+3)}$

is:

We have,

$\lim_{x\to 0}\frac{|x|\log_e(1+|\sin 2x|)}{x^2(|x|+3)}$

As $x\to 0$,

$|\sin 2x|\sim 2|x|$

Also,

$\log_e(1+|\sin 2x|)\sim |\sin 2x|$

So,

$\log_e(1+|\sin 2x|)\sim 2|x|$

Now the numerator becomes approximately:

$|x|\cdot 2|x|=2x^2$

The denominator becomes approximately:

$x^2(|x|+3)\to 3x^2$

Therefore,

$\lim_{x\to 0}\frac{|x|\log_e(1+|\sin 2x|)}{x^2(|x|+3)}=\frac{2x^2}{3x^2}$

$=\frac{2}{3}$

Hence, the limit exists and is equal to $\frac{2}{3}$.