lim-x-4-ax-b-x-x-4-3-4-a-b-

Question Number 9654 by ridwan balatif last updated on 22/Dec/16

\lim_{x \to 4} \frac{ax + b - \sqrt{x}}{x - 4} = \frac{3}{4}, a + b = \dots ?

Answered by prakash jain last updated on 22/Dec/16

\lim_{x \to 4} \frac{ax + b - \sqrt{x}}{x - 4} = \frac{3}{4}, a + b = \dots ?

Since limit exists

\lim_{x \to 4} ax + b - \sqrt{x} = 0

4 a + b - 2 = 0

b = 2 - 4 a

\frac{ax + 2 - 4 a - \sqrt{x}}{x - 4} = \frac{a (x - 4) - (\sqrt{x} - 2)}{x - 4}

= \frac{a (x - 4) - \frac{(x - 4)}{(\sqrt{x} + 2)}}{x - 4} = a - \frac{1}{\sqrt{x} + 2}

a - \frac{1}{4} = \frac{3}{4} \Rightarrow a = 1

b = 2 - 4 a = - 2

a + b = - 1

c h e c k

\lim_{x \to 4} \frac{x - 2 - \sqrt{x}}{x - 4} = \lim_{x \to 4} \frac{(x - 4) - (\sqrt{x} - 2)}{(x - 4)}

= \lim_{x \to 4} \frac{(x - 4) (1 - \frac{1}{\sqrt{x} + 2})}{(x - 4)} = \frac{3}{4}

Commented by ridwan balatif last updated on 23/Dec/16

thank you sir for your help