The limit can be written as [(sinh²x-sin²x)/x⁴(sinx/x)²(sinhx/x)²].Now,in the denominator sinx/x and sinhx/x tend to 1.So,the lim becomes

(sinhx+sinx/x)(sinhx-sinx/x³)=(2)lim(sinhx-sinx/x³)

Now apply L'hospital rule twice,we get,

(2)(sinhx+sinx/6x) which tends to (2)(2/6) as x tends to 0.Hence,the ans is 4/6=2/3