Question Number 156557 by mathlove last updated on 12/Oct/21
Commented by john_santu last updated on 13/Oct/21
$$\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{3}^{{x}} −\mathrm{1}\right)^{\mathrm{2}} \left(\mathrm{3}^{{x}} +\mathrm{1}\right)}{\mathrm{1}−\mathrm{cos}\:{x}}.\frac{\sqrt{\mathrm{2}}+\sqrt{\mathrm{1}+\mathrm{cos}\:{x}}}{\mathrm{1}} \\ $$$$=\mathrm{4}\sqrt{\mathrm{2}}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left(\mathrm{3}^{{x}} −\mathrm{1}\right)^{\mathrm{2}} }{\mathrm{2sin}\:^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)} \\ $$$$=\mathrm{2}\sqrt{\mathrm{2}}\:\left(\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{3}^{{x}} −\mathrm{1}}{{x}}.\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{x}}{\mathrm{sin}\:\frac{\mathrm{1}}{\mathrm{2}}{x}}\right)^{\mathrm{2}} \\ $$$$=\mathrm{8}\sqrt{\mathrm{2}}\:\mathrm{ln}\:^{\mathrm{2}} \left(\mathrm{3}\right) \\ $$