find a and b
lim x -> 0
((2x+b)^1/2)-(5^1/2))/ ((a(abs x)/4) - (1/4))= 2
It seems that there is a "trivial solution" by setting x=0, and solve for b after setting a=any finite number to get
b=(sqrt(5)-1/2)²
Also, to be more serious, the intention of the question seems to be that when lim x->0, both numerator and denominator become zero, thereby requiring use of the L'Hôpital's rule.
To get the numerator to become zero, b needs to be 5. Take the derivatives of the numerator and denominator, equate the quotient to 2 and solve for a.
I get a=2/√5.
Check
1. that the limits for x->0+ and x->0- to make sure they both exist and are equal to 2.
2. that by replacing |x| by +x and -x both give the same limit of 2.
Also, check my thinking and calculations.