Question

find constant a and b

Answer 1

\[\lim_{x \rightarrow 0} \frac{ \sqrt{ax+b}- 1 }{ x } \] =1

Answer 2

let the limit has 0/0
so, sqrt(ax+b) = 1 when x=0
sqrt(a(0)+b) = 1
sqrt(b) = 1
b = 1

because 0/0, use L' Hopital to find derivative of numerator and denominator,
{a/(2sqrt(ax+b))}/1 = 1 (just put x=0)
a/2(sqrt(a(0)+b)) = 1
a/2sqrt(b) = 1
a/2sqrt(1) = 1
a/2 = 1
a = 2