Compute the value of $\displaystyle \lim_{z \to 0} \left(\dfrac{1}{z^2} - \dfrac{1}{\sin^2 z}\right)$
We have used expansion, but found no leadings. Also, we try to factorize, hoping the wonders of trigonometry identity here, but seems no lead. Could you help us?
Hint:
$$\dfrac1{z^2}-\dfrac1{\sin^2z}=\dfrac{\sin z-z}{z^3}\left(\dfrac{\sin z}z+1\right)\left(\dfrac z{\sin z}\right)^2$$
Now use Are all limits solvable without L'Hôpital Rule or Series Expansion