= Solution
To retain the finite term after cancellation, expand far enough before dividing. The <Maclaurin series> are
$$
\csc x=x^{-1}+\frac x6+\frac{7x^3}{360}+O(x^5),\qquad
\sinh x=x+\frac{x^3}{6}+\frac{x^5}{120}+O(x^7).
$$
Thus
$$
\frac{\csc x}{x^3}-\frac{\sinh x}{x^5}
=\left(x^{-4}+\frac{x^{-2}}6+\frac7{360}\right)-\left(x^{-4}+\frac{x^{-2}}6+\frac1{120}\right)+O(x^2),
$$
and hence
$$
\boxed{\lim_{x\to0}\left(\frac{\csc x}{x^3}-\frac{\sinh x}{x^5}\right)=\frac1{90}.}
$$
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