If the function f ( x ) = { 72 x - 9 x - 8 x + 1 2 - 1 + cos x , x ? 0 a log e 2 log e 3 , x = 0 is continuous at x = 0 , then the value of a 2 is equal to [2024]

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Solution

Q.
If the function $f (x) = {\begin{matrix} \frac{72^{x} - 9^{x} - 8^{x} + 1}{\sqrt{2} - \sqrt{1 + \cos x}}, & x \neq 0 \\ a \log_{e} 2 \log_{e} 3, & x = 0 \end{matrix}$ is continuous at $x = 0,$ then the value of $a^{2}$ is equal to [2024]

1 746

2 968

3 1250

4 1152

Ans.
(4)

$f (x) = {\begin{matrix} \frac{72^{x} - 9^{x} - 8^{x} + 1}{\sqrt{2} - \sqrt{1 + \cos x}}, & x \neq 0 \\ a \log_{e} 2 \log_{e} 3, & x = 0 \end{matrix}$

$∵$ $f (x)$ is continuous at $x = 0$

$f (0) = \lim_{x \to 0} \frac{72^{x} - 9^{x} - 8^{x} + 1}{\sqrt{2} - \sqrt{1 + \cos x}}$

$= \lim_{x \to 0} \frac{(9^{x} - 1) (8^{x} - 1) (\sqrt{2} + \sqrt{1 + \cos x})}{\frac{(1 - \cos x)}{x^{2}} \times x^{2}}$

$= \lim_{x \to 0} (\frac{9^{x} - 1}{x}) (\frac{8^{x} - 1}{x}) \frac{(\sqrt{2} + \sqrt{1 + \cos x})}{(\frac{2 \sin^{2} \frac{x}{2}}{4 \times \frac{x^{2}}{4}})}$

$= \lim_{x \to 0} \frac{2 (\frac{9^{x} - 1}{x}) (\frac{8^{x} - 1}{x}) (\sqrt{2} + \sqrt{1 + \cos x})}{{(\frac{\sin \frac{x}{2}}{\frac{x}{2}})}^{2}}$

$= 2 \times (\log_{e} 9 \log_{e} 8 (2 \sqrt{2})) = 4 \sqrt{2} \times 2 \times 3 \log_{e} 2 \log_{e} 3$

$= 24 \sqrt{2} \log_{e} 2 \log_{e} 3 = a \log_{e} 2 \log_{e} 3$ $(∵ Given)$

$\Rightarrow a = 24 \sqrt{2}$

Hence, $a^{2} = 1152$

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