Question

$L{t}_{x\to 0}\frac{{27}^x-9^x-3^x+1}{\sqrt{2}-\sqrt{1+cosx}}=$

• A. $8\sqrt{2}(log3{)}^2$
• B. $\sqrt{2}(log3{)}^2$
• C. $16\sqrt{2}\left(log3\right)$
• D. $\left(log3\right)\left(log2\right).3\sqrt{2}$

Answer 1

The correct option is A $8\sqrt{2}(log3{)}^2$

$L{t}_{x\to 0}\frac{(9^x-1)(3^x-1)}{(1-cosx)}.[\sqrt{2}+\sqrt{1+cosx}]$
$=L{t}_{x\to 0}\frac{\frac{a^x-1}{x}.\frac{3^x-1}{x}}{\frac{si{n}^{2}x}{x^2}}.(\sqrt{2}+\sqrt{1+cosx})(1+cosx)$
$=\frac{log{3}^2.log3}{1^2}.(\sqrt{2}+\sqrt{2})(1+1)$
$=8\sqrt{2}(log3{)}^2$