Evaluate the limit \[ \lim _{x \rightarrow 0} \frac{\sinh (3 x)-\sin (3 x)}{\sin \left(x^{3}\right)} \]

Step 1 ：Simplify the expression by factoring out a common factor of 3x in the numerator: \(\lim _{x \rightarrow 0} \frac{3x(\cosh (3 x)-\sin (3 x))}{\sin \left(x^{3}\right)}\)

Step 2 ：Apply L'Hôpital's rule by taking the derivative of the numerator and denominator with respect to x:

Step 3 ：Derivative of the numerator: \(\frac{d}{dx} (3x(\cosh (3 x)-\sin (3 x))) = 3(\cosh (3 x)-\sin (3 x)) + 3x(3\sinh (3 x)-3\cos (3 x))\)

Step 4 ：Derivative of the denominator: \(\frac{d}{dx} (\sin \left(x^{3}\right)) = 3x^2\cos(x^3)\)

Step 5 ：Evaluate the limit of the derivative of the expression: \(\lim _{x \rightarrow 0} \frac{3(\cosh (3 x)-\sin (3 x)) + 3x(3\sinh (3 x)-3\cos (3 x))}{3x^2\cos(x^3)}\)

Step 6 ：Plugging in x=0, we get: \(\frac{3(\cosh (0)-\sin (0)) + 0(3\sinh (0)-3\cos (0))}{0^2\cos(0^3)} = \frac{3(1-0)}{0} = \infty\)

Step 7 ：Therefore, the limit of the given expression as x approaches 0 is \(\boxed{\infty}\)