If-lim-x-0-ax-1-2x-1-3-1-bx-1-1-2-then-a-b-

Question Number 161002 by bobhans last updated on 10/Dec/21

If \lim_{x \to 0} \frac{ax}{\sqrt[3]{1 + 2 x} \sqrt{1 + bx} - 1} = \frac{1}{2}

then a \times b = ?

Answered by blackmamba last updated on 11/Dec/21

\lim_{x \to 0} \frac{a x}{(1 + \frac{2 x}{3}) (1 + \frac{b x}{2}) - 1} = \frac{1}{2}

\lim_{x \to 0} \frac{a x}{(\frac{2}{3} + \frac{b}{2}) x + \frac{2 {b x}^{2}}{6}} = \frac{1}{2}

\frac{6 a}{4 + 3 b} = \frac{1}{2} \Rightarrow a = \frac{4 + 3 b}{12}

a \times b = \frac{4 b + 3 b^{2}}{12}