Find the following limit as x->0

1. Sep 18, 2005

thenewbosco

find the following limit as x-->0

Hello I am trying to find the following limit as x-->0. I have tried using l'hopital's rule but all it produces is a more complex thing i still can't take the limit of. any help?
$$\frac{sin x - x}{sin^3(x)}$$
I can see this limit exists on the graph i just dont know how to go about solving it.

2. Sep 18, 2005

arildno

Take L'hopital once.
Then multiply your expression with $$1=\frac{\cos(x)+1}{\cos(x)+1}$$

3. Sep 18, 2005

thenewbosco

Thanks this worked. One question about this is how did you know to multiply by this? How would one know to do this when faced with similar situations

4. Sep 18, 2005

Hurkyl

Staff Emeritus
You could have just done L'Hôpital again.

arildno just factored the denominator (after converting sin² to cos²).

5. Sep 18, 2005

arildno

Just practice&patience.
To think of simple tricks like this comes naturally to you if you have worked assiduously with your maths earlier.

6. Sep 18, 2005

arildno

Factorization is more fun.