I am having trouble with this limit. I don't need an answer, but I would appreciate if someone could give me a hint or show me what I am missing.

The following is what I can make sense of it so far:

$\displaystyle \lim_{x \to 0}\frac{tan\theta - sin\theta}{sin^3\theta}$

$\displaystyle \rightarrow \lim_{x \to 0}\frac{\frac{sin\theta}{cos\theta} - sin\theta}{sin^3\theta}$

$\displaystyle \rightarrow \lim_{x \to 0}\left(\frac{sin\theta}{cos\theta}\right)\left( \frac{1}{sin^3\theta}\right) - \frac{sin\theta}{sin^3\theta}$

$\displaystyle \rightarrow \lim_{x \to 0}\frac{1}{cos\theta cos^2\theta} - \frac{1}{sin^2\theta}$

$\displaystyle \rightarrow \lim_{x \to 0}\frac{1- cos\theta}{cos\theta sin^2\theta}$

I am aware that $\displaystyle \lim_{x \to 0}\frac{1- cos x}{x} = 0$; however, I am a little lost on how to get there, or if that is even the direction I should be taking. Any hints or explanations would be greatly appreciated. Thanks!

Take care.

/alan