Limit of Ratio of Successive Terms for Generalized Fibonacci Sequence

Th Fibonacci sequence is generated by the rule  
\[a_{n+1} = a_n + a_{n-1}\]
  with  
\[a_0 =a_1 =1\]
. In English we add two terms to get the next term.
The ratio of successive Fibonacci numbers tends to a limit  
\[\frac{a_{n+1}}{a+n} = \frac{ \sqrt{5} -1}{2}\]
  as  
\[n \rightarrow \infty\]
  Consider the 'generalized Fibonacci sequence' defined by the rule  
\[a_{n+1} =A a_n +B a_{n-1} , \: A, B > 0\]

Does the ration of successive terms to to a limit for this sequence? Yes it does. Divide the rule that generates this sequence by  
\[a_n\]
  to give
\[\frac{a_{n+1}}{a_n} =A +B \frac{a_{n-1}}{a_n}\]

As
\[n \rightarrow \infty\]
, if
\[\frac{a_{n+1}}{a_n} \rightarrow l\]
  then
\[\frac{a_n}{a_{n-1}} \rightarrow \frac{1}{l}\]

Then  
\[\frac{a_{n+1}}{a_n} =A +B \frac{a_{n-1}}{a_n}\]
  becomes
\[l =A +\frac{B}{l}\]

Multiply by  
\[l\]
  and rearrange to obtain  
\[l^2 - Al -B=0\]

The solutions are  
\[l= \frac{A \pm \sqrt{A^2 +4B}}{2}\]

The positive option gives the correct value of  
\[l\]
. The negative option gives a negative vale. This is wrong since  
\[A,B >0\]
  so  
\[A .