Mathematics - Continuity - Differentiability Question with Solution | TestHub

$\begin{array}{rl}& \underset{x\to 0}{\lim }f(x)=a\ln 2\ln 3\\ & \underset{n\to 0}{\lim }\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}}=\underset{x\to 0}{\lim }\frac{\left(8^x-1\right)\left(9^x-1\right)}{\sqrt{2}-\sqrt{1+\cos x}}\\ & \underset{n\to 0}{\lim }\left(\frac{8^x-1}{x}\right)\left(\frac{9^x-1}{x}\right)\left(\frac{x^2}{1-\cos x}\right)(\sqrt{2}+\sqrt{1+\cos x})\\ & \therefore \ln 8\times \ln 9\times 2\times 2\sqrt{2}=24\sqrt{2}\ln 2\ln 3\\ & \therefore a=24\sqrt{2},a^2=576\times 2=1152\end{array}$