Limit of x^2/(1-cosx) as x→0 - Solution without LH Rule

[Frillth]

Homework Statement

Find the limit as x approaches 0 of x^2/(1-cosx).

Homework Equations

None.

The Attempt at a Solution

I know from L'Hopital's rule that the limit is 2, but I'm not supposed to use L'Hopital's rule to calculate it. What else can I do here?

 

[morphism]

How about the identity cos(x) = 1 - 2sin^2(x/2)?

 

[Dick]

You could also consider the power series expansion of cos(x) around zero.

 

[HallsofIvy]

Multiply both numerator and denominator by 1+ cos(x). Then use the fact that
$$lim_{x\rightarrow 0}\frac{x}{sin(x)}= 1$$.