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  4. displaystyle limx→2 (3x+33-x-12/3-x/2 - 31-x) is equal to .

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Q. $\displaystyle\lim_{x\to2} \frac{3^{x}+3^{3-x}-12}{3^{-x/2} - 3^{1-x}}$ is equal to __________.

JEE MainJEE Main 2020Limits and Derivatives

Solution:

$lim_{x \to 2} \frac{3^{x}+3^{3-x}-12}{3^{-x/2} - 3^{1-x}} \Rightarrow lim_{x \to 2} \frac{3^{2x}-12.3^{x}+27}{3^{x/2} - 3}$
$= lim_{x \to 2} \frac{\left(3^{x}-9\right)\left(3^{x}-3\right)}{\left(3^{x/2} - 3\right)}$
$= lim_{x \to 2} \frac{\left(3^{x/2} + 3\right)\left(3^{x/2} - 3\right)\left(3^{x} - 3\right)}{\left(3^{x/2} - 3\right)}$
$= 36$