lim _{x rightarrow 2} frac{3^x+3^{3-x}-12}{3^ | vedclass

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Question

(A) Let $f(x) = \frac{3^x + 3^{3-x} - 12}{3^{3-x} - 3^{x/2}}$.Multiply the numerator and denominator by $3^x$:$\lim _{x \rightarrow 2} \frac{3^{2x} -

Vedclass · Accepted Answer

$-\frac{4}{3}$

Vedclass · Answer

(A) Let $f(x) = \frac{3^x + 3^{3-x} - 12}{3^{3-x} - 3^{x/2}}$.Multiply the numerator and denominator by $3^x$:$\lim _{x \rightarrow 2} \frac{3^{2x} - 12 \cdot 3^x + 27}{3^3 - 3^{3x/2}}$Let $t = 3^{x/2}$. As $x \rightarrow 2$,$t \rightarrow 3$.The expression becomes $\lim _{t \rightarrow 3} \frac{t^4 - 12t^2 + 27}{27 - t^3}$.Factor the numerator: $t^4 - 12t^2 + 27 = (t^2 - 9)(t^2 - 3) = (t-3)(t+3)(t^2-3)$.Factor the denominator: $27 - t^3 = (3-t)(9 + 3t + t^2) = -(t-3)(t^2 + 3t + 9)$.$\lim _{t \rightarrow 3} \frac{(t-3)(t+3)(t^2-3)}{-(t-3)(t^2 + 3t + 9)} = \lim _{t \rightarrow 3} \frac{-(t+3)(t^2-3)}{t^2 + 3t + 9}$.Substitute $t=3$: $\frac{-(3+3)(9-3)}{9+9+9} = \frac{-6 \cdot 6}{27} = \frac{-36}{27} = -\frac{4}{3}$.

$\lim _{x \rightarrow 2} \frac{3^x+3^{3-x}-12}{3^{3-x}-3^{\frac{x}{2}}} = $

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