lim _{x rightarrow 3 / 2} frac{left(4 x^2-6 x | vedclass

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(A) $	ext{Given limit: } L = \lim _{x ightarrow \frac{3}{2}} \frac{2x(2x-3)(4x^2+6x+9)}{(2x)^{1/3} - 3^{1/3}}$$	ext{Rationalizing the denominator

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$\sqrt[3]{3^{17}}$

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(A) $	ext{Given limit: } L = \lim _{x ightarrow \frac{3}{2}} \frac{2x(2x-3)(4x^2+6x+9)}{(2x)^{1/3} - 3^{1/3}}$$	ext{Rationalizing the denominator using } a^3 - b^3 = (a-b)(a^2+ab+b^2) 	ext{ where } a=(2x)^{1/3}, b=3^{1/3}:$$L = \lim _{x ightarrow \frac{3}{2}} \frac{2x(2x-3)(4x^2+6x+9)((2x)^{2/3} + (6x)^{1/3} + 3^{2/3})}{2x-3}$$L = \lim _{x ightarrow \frac{3}{2}} 2x(4x^2+6x+9)((2x)^{2/3} + (6x)^{1/3} + 3^{2/3})$$	ext{Substituting } x = \frac{3}{2}:$$L = 2(\frac{3}{2})(4(\frac{9}{4}) + 6(\frac{3}{2}) + 9)(3^{2/3} + 9^{1/3} + 3^{2/3})$$L = 3(9 + 9 + 9)(3 	imes 3^{2/3})$$L = 3(27)(3^{5/3}) = 3^1 	imes 3^3 	imes 3^{5/3} = 3^{1+3+5/3} = 3^{17/3} = \sqrt[3]{3^{17}}$

$\lim _{x \rightarrow 3 / 2} \frac{\left(4 x^2-6 x\right)\left(4 x^2+6 x+9\right)}{\sqrt[3]{2 x}-\sqrt[3]{3}}=$

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