If f(x) = \ array ll ( 2 ^ x - 1) ^ 2 3x xsi n ^ 2 x λ array array...

$∵ f$\left(x\right)$is continuous at$x = 0$<br />$∴ \lim_{x \to 0}f\left(x\right) = f\left(0\right)$<br />$∵ \lim_{x \to {0}^{+}}\dfrac{{\left({2}^{x} - 1\right)}^{2}\tan 3x}{x . \sin^{2}x} = \lambda$<br />$∵ \lim_{x \to {0}^{+}}{\left(\dfrac{{2}^{x} - 1}{x}\right)}^{2}\dfrac{\tan 3x}{\left(3x\right)} . \dfrac{\left(3x\right)}{x}\dfrac{{x}^{2}}{si{n}^{2}x} = \lambda$<br />${\left(\ln 2\right)}^{2} . 1.3 . 1 = \lambda$<br />$\dfrac{10\sqrt{3\lambda}}{\ln 2} = 30