lim (1/x² − 1/sin² x)

Common denominator.
$$\frac1{x^2}-\frac1{\sin^2x}=\frac{\sin^2x-x^2}{x^2\sin^2x}.$$

Numerator. $\sin x=x-\frac{x^3}6+o(x^3)$, so
$$\sin^2x=x^2-\frac{x^4}3+o(x^4)\quad\Longrightarrow\quad\sin^2x-x^2=-\frac{x^4}3+o(x^4).$$

Denominator. $x^2\sin^2x=x^4+o(x^4)$.

Limit.
$$\frac{-\frac{x^4}3+o(x^4)}{x^4+o(x^4)}\longrightarrow-\frac13.$$

Remark. The orders of expansion are dictated by the denominator: $x^4$ below means the numerator must be known up to $o(x^4)$, so we need $\sin x$ up to $x^3$ only.