Let a, b ∈ R, b ≠ 0 . Defined a function, f(x) = \ array ll...

We have,

f\left(x\right) = \left\{$\begin{array}{ll}asin\dfrac{\pi}{2}\left(x - 1\right),forx \le 0 \\ \dfrac{\tan 2x - \sin 2x}{b{x}^{3}}, forx > 0\end{array}$\right)</p><p>If function is continuous atx = 0$, then</p><p>$LHL = RHL = f\left(0\right)$</p><p>Now,</p><p>$LHL = \lim_{x \to {0}^{-}}\left[asin\dfrac{\pi}{2}\left(x - 1\right)\right]$</p><p>$⇒ LHL = \lim_{h \to 0}\left[asin\dfrac{\pi}{2}\left(0 - h - 1\right)\right]$</p><p>$⇒ LHL = \lim_{h \to 0}\left[asin\dfrac{\pi}{2}\left(- 1\right)\right]$</p><p>$⇒ LHL = - a$</p><p>Now,</p><p>$RHL = \lim_{x \to {0}^{+}}\left(\dfrac{\tan 2x - \sin 2x}{b{x}^{3}}\right)$</p><p>$∵ \sin x = x - \dfrac{{x}^{3}}{3 !} + \dfrac{{x}^{5}}{5 !} + \ldots \ldots$</p><p>$∵ \tan x = x + \dfrac{{x}^{3}}{3} + \dfrac{2{x}^{5}}{15} + \ldots .$</p><p>$RHL = \lim_{x \to {0}^{+}}\left[\dfrac{\dfrac{( 2x{)}^{3}}{3} + \dfrac{( 2x{)}^{3}}{6}}{b{x}^{3}}\right] = \dfrac{\dfrac{8}{3} + \dfrac{8}{6}}{b} = \dfrac{4}{b}$</p><p>And,</p><p>$f ( 0 ) = asin\left[\dfrac{\pi}{2} ( 0 - 1 )\right] = asin\left(- \dfrac{\pi}{2}\right) = - a$</p><p>Then,</p><p>$RHL = f\left(0\right)$</p><p>$⇒ \dfrac{4}{b} = - a ⇒ - ab = 4 ⇒ 10 - ab = 14