If-lim-x-3-3x-7-20x-4-1-3-ax-b-x-3-2-exist-what-is-the-value-of-ab-

Question Number 95980 by john santu last updated on 29/May/20

If \lim_{x \to 3} \frac{\sqrt{3 x + 7} - \sqrt[3]{20 x + 4} + a x + b}{{(x - 3)}^{2}}

exist, what is the value of a b

Answered by i jagooll last updated on 29/May/20

set x - 3 = y, x = y + 3, y \to 0

\lim_{y \to 0} \frac{\sqrt{3 y + 16} - \sqrt[3]{20 y + 64} + a y + 3 a + b}{y^{2}}

n u m e r a t o r m u s t b e = 0

\Rightarrow 4 - 4 + 3 a + b = 0, b = - 3 a

L^{'} Hopital

\lim_{y \to 0} \frac{\frac{3}{2 \sqrt{3 y + 16}} - \frac{20}{3 \sqrt[3]{{(20 y + 64)}^{2}}} + a}{2 y}

\Rightarrow numerator must be = 0

\Rightarrow \frac{3}{8} - \frac{20}{3.16} + a = 0

a = \frac{5}{12} - \frac{3}{8} = \frac{10 - 9}{24} = \frac{1}{24}

t h e n b = - 3 a = - \frac{1}{8}

s o a \times b = - \frac{1}{192} . ∙

Commented by john santu last updated on 29/May/20

greatt