lim _{x rightarrow 0} frac{27^x-9^x-3^x+1}{sq | vedclass

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(C) Let $L = \lim _{x \rightarrow 0} \frac{27^x-9^x-3^x+1}{\sqrt{5}-\sqrt{4+\cos x}}$.Factor the numerator: $27^x-9^x-3^x+1 = 9^x(3^x-1) - 1(3^x-1) =

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$8 \sqrt{5}(\log 3)^2$

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(C) Let $L = \lim _{x \rightarrow 0} \frac{27^x-9^x-3^x+1}{\sqrt{5}-\sqrt{4+\cos x}}$.Factor the numerator: $27^x-9^x-3^x+1 = 9^x(3^x-1) - 1(3^x-1) = (9^x-1)(3^x-1)$.Rationalize the denominator: $\frac{1}{\sqrt{5}-\sqrt{4+\cos x}} = \frac{\sqrt{5}+\sqrt{4+\cos x}}{5-(4+\cos x)} = \frac{\sqrt{5}+\sqrt{4+\cos x}}{1-\cos x}$.So,$L = \lim _{x \rightarrow 0} \frac{(9^x-1)(3^x-1)}{1-\cos x} \cdot (\sqrt{5}+\sqrt{4+\cos x})$.Using $1-\cos x = 2 \sin^2(x/2)$,we have $L = \lim _{x \rightarrow 0} \frac{(9^x-1)(3^x-1)}{2 \sin^2(x/2)} \cdot (\sqrt{5}+\sqrt{4+\cos x})$.Multiply and divide by $x^2$: $L = \lim _{x}$ ${\rightarrow 0} \frac{(\frac{9^x-1}{x})(\frac{3^x-1}{x})}{2 \cdot (\frac{\sin(x/2)}{x/2})^2 \cdot (1/4)} \cdot (\sqrt{5}+\sqrt{4+\cos x})$.$L = \frac{\ln 9 \cdot \ln 3}{2 \cdot 1 \cdot (1/4)} \cdot (\sqrt{5}+\sqrt{5}) = \frac{2 \ln 3 \cdot \ln 3}{1/2} \cdot 2 \sqrt{5} = 8 \sqrt{5} (\ln 3)^2$.

$\lim _{x \rightarrow 0} \frac{27^x-9^x-3^x+1}{\sqrt{5}-\sqrt{4+\cos x}}=$

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