l i m x → 2 3 x + 3 3 - x - 12 3 - x 2 - 3 1 - x is equal to

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\underset{x \to 2}{l i m} \frac{3^{x} + 3^{3 - x} - 12}{3^{- \frac{x}{2}} - 3^{1 - x}}

is equal to

MathematicsLimitsJEE MainJEE Main 2020 (07 Jan Shift 1)

Solution:

1981 Upvotes Verified Answer

The correct answer is: 36

Let, $3^{\frac{x}{2}} = t, x \to 2 \Rightarrow t \to 3$

$\lim_{t \to 3} \frac{t^{2} + \frac{27}{t^{2}} - 12}{\frac{1}{t} - \frac{3}{t^{2}}} = \lim_{t \to 3} \frac{t^{4} + 27 - 12 t^{2}}{t - 3}$

$\lim_{t \to 3} \frac{(t^{2} - 3) (t + 3) (t - 3)}{(t - 3)} = 6 \times 6 = 36$

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