How do we find the value of a without using L'hospitals rule?

lim x-->0 [(1+ax) - (1+x)^(1/2)]/(x^2) } =(1/8)

Hmmm. Using L'hospitals rule, we have the value as

(a - 1/2 (1+x)^(-1/2))/2x
(1/4 (1+x)^(-3/2)/2
-> 1/8

So, I don't see how we can assign a value to a, because the limit is 1/8 for any value of a.

Am I missing something?