A) \[4\sqrt{2}{{(log3)}^{2}}\]
B) \[8\sqrt{2}{{(log3)}^{2}}\]
C) \[2\sqrt{2}{{(log3)}^{2}}\]
D) None of these
Correct Answer: B
Solution :
[b] \[\underset{x\to 0}{\mathop{\lim }}\,\frac{{{27}^{x}}-{{9}^{x}}-{{3}^{x}}+1}{\sqrt{2}-\sqrt{1+\cos x}}\] \[=\underset{x\to 0}{\mathop{\lim }}\,\frac{{{9}^{x}}{{.3}^{x}}-{{9}^{x}}-{{3}^{x}}+1}{\sqrt{2}-\sqrt{2}\cos \frac{x}{2}}\] \[=\underset{x\to 0}{\mathop{\lim }}\,\left( \frac{{{9}^{x}}-1}{x} \right).\left( \frac{{{3}^{x}}-1}{x} \right).\frac{1}{\sqrt{2}}.{{x}^{2}}.\frac{1}{2{{\sin }^{2}}\frac{x}{4}}\] \[=\underset{x\to 0}{\mathop{\lim }}\,\left( \frac{{{9}^{x}}-1}{x} \right).\left( \frac{{{3}^{x}}-1}{x} \right).\frac{1}{\sqrt{2}}\left( \frac{{{x}^{2}}/16}{{{\sin }^{2}}x/4} \right)8\] \[=\frac{8}{\sqrt{2}}(log9)(log3)=8\sqrt{2}{{(log3)}^{2}}.\]
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