<s>[INST] $\lim \limits_{x \to 0} \frac{1}{x^2}-\csc^2x $ using L'Hôpital's rule This is from an exercise set on L'Hôpital's rule. I've never seen anything so stubborn and recalcitrant. Anybody have suggestions?
$\lim \limits_{x \to 0} \frac{1}{x^2}-\csc^2x $ 
 [/INST] As $x\to0$,
$$\frac{1}{x^2}-\frac{1}{\sin^2 x}=\frac{1}{x^2}-\frac{1}{(x-x^3/6+o(x^3))^2}=\frac{1}{x^2}-\frac{1}{x^2-x^4/3+o(x^4)}\\=\frac{1}{x^2}-\frac{1}{x^2}\left(\frac{1}{1-x^2/3+o(x^2)}\right)=\frac{1}{x^2}-\frac{1}{x^2}\left(1+x^2/3+o(x^2)\right)=-\frac{1}{3}+o(1)\to-\frac{1}{3}$$
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