If the function f(x)= ll 72^x-9^x-8^x+1 2-1+ x & , x 0 a _e 2 _c … - Vidyadip

MCQ

If the function $f(x)=\left\{\begin{array}{ll}\frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}} & , x \neq 0 \\ a \log _e 2 \log _c 3 & , x=0\end{array}\right.$ is continuous at $x = 0,$ then the value of $a ^2$ is equal to

A
968
B
1152
C
746
D
1250

✓

Answer

$\lim _{x \rightarrow 0} f(x)=a \ln 2 \ln 3$
$\lim _{n \rightarrow 0} \frac{72^x-9^x-8^x+1}{\sqrt{2}-\sqrt{1+\cos x}}$
$=\lim _{x \rightarrow 0} \frac{\left(8^x-1\right)\left(9^x-1\right)}{\sqrt{2}-\sqrt{1+\cos x}}$
$\lim _{n \rightarrow 0}\left(\frac{8^x-1}{x}\right)\left(\frac{9^x-1}{x}\right)\left(\frac{x^2}{1-\cos x}\right)(\sqrt{2}+\sqrt{1+\cos x})$
$\therefore \ln 8 \times \ln 9 \times 2 \times 2 \sqrt{2}$
$=24 \sqrt{2} \ln 2 \ln 3$
$\therefore a=24 \sqrt{2}, a^2$
$=576 \times 2=1152$

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