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lim x-2 3^x +3^3-x -12 / 3^{3-x -}3^x/2

$\lim_{x \to 2} \frac{3^{x} + 3^{3 - x} - 12}{3^{3 - x} - 3^{\frac{x}{2}}} = \lim_{x \to 2} \frac{3^{x} + 3^{3} . 3^{- x} - 12}{3^{3} . 3^{- x} - 3^{\frac{x}{2}}} = \lim_{x \to 2} \frac{3^{x} + \frac{3^{3}}{3^{x}} - 12}{\frac{3^{3}}{3^{x}} - 3^{\frac{x}{2}}} = \lim_{x \to 2} \frac{\frac{3^{x} . 3^{x} + 27 - 12. 3^{x}}{3^{x}}}{\frac{3^{3} - 3^{\frac{x}{2}} . 3^{x}}{3^{x}}} = \lim_{x \to 2} \frac{{(3^{x})}^{2} - 12. 3^{x} + 27}{3^{3} - 3^{\frac{x}{2} + x}} = \lim_{x \to 2} \frac{{(3^{x})}^{2} - 9. 3^{x} - 3. 3^{x} + 27}{3^{3} - 3^{\frac{3 x}{2}}} = \lim_{x \to 2} \frac{3^{x} (3^{x} - 9) - 3 (3^{x} - 9)}{3^{3} - {(3^{\frac{x}{2}})}^{3}} = \lim_{x \to 2} \frac{3^{x} (3^{x} - 9) - 3 (3^{x} - 9)}{3^{3} - {(3^{\frac{x}{2}})}^{3}} = \lim_{x \to 2} \frac{(3^{x} - 3) (3^{x} - 9)}{(3 - 3^{\frac{x}{2}}) (3^{2} + 3. 3^{\frac{x}{2}} + {(3^{\frac{x}{2}})}^{2})} = \lim_{x \to 2} \frac{(3^{x} - 3) ({(3^{\frac{x}{2}})}^{2} - 3^{2})}{- (3^{\frac{x}{2}} - 3) (9 + 3^{\frac{x}{2} + 1} + 3^{x})} = - \lim_{x \to 2} \frac{(3^{x} - 3) (3^{\frac{x}{2}} + 3) (3^{\frac{x}{2}} - 3)}{(3^{\frac{x}{2}} - 3) (9 + 3^{\frac{x}{2} + 1} + 3^{x})} = - \lim_{x \to 2} \frac{(3^{x} - 3) (3^{\frac{x}{2}} + 3)}{9 + 3^{\frac{x}{2} + 1} + 3^{x}} = - \frac{(3^{2} - 3) (3^{\frac{2}{2}} + 3)}{9 + 3^{\frac{2}{2} + 1} + 3^{2}} = - \frac{(9 - 3) (3 + 3)}{9 + 9 + 9} = - \frac{6 \times 6}{27} = - \frac{36}{27} = - \frac{4}{3}$

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