Limx→027x−9x−3x+12−1+cosx=

Limx→0(9.3)x−9x−3x+12−1+cosx×2+1+cosx2+1+cosx=Limx→0(9x−1)(3x−1)x2x21−cosx(2+1+cosx)(limx→0ax−1x=loga,limx→01−cosxx2=12)

Limx→027x−9x−3x+12−1+cosx=

Limx→0(9.3)x−9x−3x+12−1+cosx×2+1+cosx2+1+cosx=Limx→0(9x−1)(3x−1)x2x21−cosx(2+1+cosx)(limx→0ax−1x=loga,limx→01−cosxx2=12)

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$\underset{x \to 0}{L i m} \frac{27^{x} - 9^{x} - 3^{x} + 1}{\sqrt{2} - \sqrt{1 + \cos x}} =$

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

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

answer is B.

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Detailed Solution

$\underset{x \to 0}{L i m} \frac{{(9.3)}^{x} - 9^{x} - 3^{x} + 1}{\sqrt{2} - \sqrt{1 + \cos x}} \times \frac{\sqrt{2} + \sqrt{1 + \cos x}}{\sqrt{2} + \sqrt{1 + \cos x}}$ $= \underset{x \to 0}{L i m} \frac{(9^{x} - 1) (3^{x} - 1)}{x^{2}} \frac{x^{2}}{1 - \cos x} (\sqrt{2} + \sqrt{1 + \cos x})$ $(\underset{x \to 0}{l i m} \frac{a^{x} - 1}{x} = \log a, \lim_{x \to 0} \frac{1 - \cos x}{x^{2}} = \frac{1}{2})$