lim-x-1-ax-a-b-3-x-1-3-2-find-a-and-b-without-L-hopital-rule-

Question Number 82699 by jagoll last updated on 23/Feb/20

\lim_{x \to 1} \frac{\sqrt{a x - a + b} - 3}{x - 1} = - \frac{3}{2}

f i n d a a n d b w i t h o u t L^{'} h o p i t a l r u l e

Commented by john santu last updated on 23/Feb/20

(1) l i m i t m u s t b e \frac{0}{0}

\Rightarrow \sqrt{b} = 3, b = 9

(2) \lim_{x \to 1} \frac{\sqrt{a (x - 1) + 9} - 3}{x - 1} = - \frac{3}{2}

l e t x - 1 = t \Rightarrow \lim_{t \to 0} \frac{\sqrt{a t + 9} - 3}{t} = - \frac{3}{2}

\lim_{t \to 0} \frac{a t}{t} \times \lim_{t \to 0} \frac{1}{\sqrt{a t + 9} + 3} = - \frac{3}{2}

a \times \frac{1}{6} = - \frac{3}{2} \Rightarrow a = - 9

Commented by jagoll last updated on 23/Feb/20

t h a n k s

Commented by mathmax by abdo last updated on 23/Feb/20

\Rightarrow {l i m}_{x \to 1} \frac{\sqrt{a x - a + b} - 3}{x - 1} + \frac{3}{2} = 0 \Rightarrow

{l i m}_{x \to 1} \frac{2 \sqrt{a x - a + b} - 6 + 3 x - 3}{2 (x - 1)} = 0 \Rightarrow

{l i m}_{x \to 1} u (x) = 0 a n d {l i m}_{x \to 1} u^{^{'}} (x) = 0 w i t h

u (x) = 2 \sqrt{a x - a + b} + 3 x - 9

{l i m}_{x \to 1} u (x) = 0 \Rightarrow 2 \sqrt{b} - 6 = 0 \Rightarrow \sqrt{b} = 3 \Rightarrow b = 9

u^{^{'}} (x) = \frac{a}{\sqrt{a x - a + b}} + 3

{l i m}_{x \to 1} u^{^{'}} (x) = 0 \Rightarrow \frac{a}{\sqrt{b}} + 3 = 0 \Rightarrow \frac{a}{3} + 3 = 0 \Rightarrow \frac{a}{3} = - 3 \Rightarrow a = - 9