By definition, .

Assume the . This implies and thus . We can thus recursively(or reverse inductively) claim that . Proceeding until n=2, we find that our hypothesis implies

Thus for all positive integers n.

By definition,

If youprovethat the limit exists, finding it can be easy.

So by definition, .

Let the limit of the ratio be L, then

So .

So solve the quadratic and choose the appropriate root.