QJEEM24Ma4f4a: JEE Main 2024 Mathematics - Class 12 JEEM24Ma4f4a Solution: If the function is continuous at , then the value of is equal to

$$f(x)=\{\begin{array}{c}\frac{{72}^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}},x\ne 0\hfill \\ a{\log }_{e}2{\log }_{e}3,x=0\hfill \end{array}$$

$\because f(x)$ is continuous at $x=0$

$\begin{aligned}& \Rightarrow \lim _{x \rightarrow 0} \frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}} \\& \lim _{x \rightarrow 0} \frac{\left(9^x-1\right)\left(8^x-1\right)(\sqrt{2}+\sqrt{1+\cos x})}{\frac{(1-\cos x)}{x^2} \times x^2} \\& =(\ln 9 \cdot \ln 8)(2 \sqrt{2}) \times 2 \\& =4 \sqrt{2} \times 2 \times 3 \ln 2 \cdot \ln 3 \\& 24 \sqrt{2} \cdot \ln 2 \cdot \ln 3 \\& \Rightarrow \quad a=24 \sqrt{2} \\& \quad a^2=1152\end{aligned}$