$\lim _{x \rightarrow 2} \frac{3^{x}+3^{3-x}-12}{3^{-x / 2}-3^{1-x}}$ is equal to \_\_\_\_

JEE Main · 2020 · 07 Jan 2020 · Shift 1 · Limits

lim_{x \to 2} \frac{3 ^{x} + 3 ^{3 - x} - 12}{3 ^{- x /2} - 3 ^{1 - x}}

is equal to ____

Answer: 36

Let

t = 3^{x /2}

, so as

x \to 2

t \to 3

. Then

3^{x} = t^{2}

3^{3 - x} = \frac{27}{t ^{2}}

3^{- x /2} = \frac{1}{t}

and

3^{1 - x} = \frac{3}{t ^{2}}

, and the expression becomes

\frac{t ^{4} - 12 t ^{2} + 27}{t - 3}

. Factoring,

t^{4} - 12 t^{2} + 27 = (t^{2} - 3) (t^{2} - 9) = (t^{2} - 3) (t - 3) (t + 3)

, so the limit is

lim_{t \to 3} (t^{2} - 3) (t + 3) = (9 - 3) (6) = 36

JEE Main · 2020 · 07 Jan 2020 · Shift 1 · Integer answer

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