If $\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{l}\frac{{36}^{\mathrm{x}}-9^{\mathrm{x}}-4^{\mathrm{x}}+1}{\sqrt{2}-\sqrt{(1+\cos \mathrm{x})}},\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{x}\ne 0\\ \mathrm{\lambda },\hspace{0.25em} \hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\mathrm{x}=0\end{array}\right.$is continuous at x = 0, then $\mathrm{\lambda }=\sqrt{\mathrm{\mu }}\ln 2\cdot \ln 3$ then the value of $\mathrm{\mu }$ must be  

f is continuous at x = 0                 

                          

$\begin{array}{l}\Rightarrow f\left(0\right)=\mathrm{\lambda }=\underset{\mathrm{h}\to 0}{lim}f\left(0+h\right)= \underset{\mathrm{h}\to 0}{lim} \left\{\frac{{36}^{\mathrm{h}}-9^{\mathrm{h}}-4^{\mathrm{h}}+1}{\sqrt{2}-\sqrt{(1+\cos \mathrm{h})}}\right\}\\ \\ =\underset{\mathrm{h}\to 0}{lim} \frac{\left(9^{\mathrm{h}}-1\right)\left(4^{\mathrm{h}}-1\right)(\sqrt{2}+\sqrt{(1+\cosh )}}{(\sqrt{2}-\sqrt{1+\cosh })(\sqrt{2}+\sqrt{1+\cosh )}}\\ =\underset{\mathrm{h}\to 0}{lim} \frac{\left(9^{\mathrm{h}}-1\right)\left(4^{\mathrm{h}}-1\right)(\sqrt{2}+\sqrt{(1+\cos \mathrm{h}})}{(1-\cos \mathrm{h})}\\ =\underset{\mathrm{h}\to 0}{lim} \frac{\left(\frac{9^{\mathrm{h}}-1}{\mathrm{h}}\right)\left(\frac{4^{\mathrm{h}}-1}{\mathrm{h}}\right)(\sqrt{2}+\sqrt{(1+\cos \mathrm{h})}}{\left(\frac{1-\cos \mathrm{h}}{{\mathrm{h}}^2}\right)}\\ =\frac{\ln 9\cdot \ln 4\cdot 2\sqrt{2}}{\left(\frac{1}{2}\right)}\end{array}$

$\begin{array}{l}=16\sqrt{2}\ln 2\cdot \ln 3\\ =\sqrt{\left(512\right)}\cdot \ln 2\cdot \ln 3\\ \mathrm{\mu }=512\end{array}$