Find numbers a and b such that:

Question

lim (sqrt(ax+b)-9)/x =1
x->0

Answer 1

lim (√(ax+b)-9)/x
= lim a/(2√(ax+b))

so, since we afre clearly dealing with lim = 0/0, we need

√(ax+b) = 9
ax+b = 81

a/(2√(ax+b)) = 1
a = 2√(ax+b)
a^2 = 4(ax+b)

a^2 = 4*81
a = 18
so, b=81

lim (√(18x+81)-9)/x = 1

verify at

https://www.wolframalpha.com/input/?i=limit+%28x-%3E0%29+%28%E2%88%9A%2818x%2B81%29-9%29%2Fx