Limx→∞a1/x+b1/x+c1/x3x=

limx→∞a1/x+b1/x+c1/x3x=elimx→∞xa1/x+b1/x+c1/x3−1=elimx→∞xa1/x+b1/x+c1/x−33=elimx→∞xa1/x−1+b1/x−1+c1/x−13=elimx→∞13a1/x−11x+b1/x-11x+c1/x-11x=elimx→∞13loga+logb+logc=e13logabc=abc13

Limx→∞a1/x+b1/x+c1/x3x=

limx→∞a1/x+b1/x+c1/x3x=elimx→∞xa1/x+b1/x+c1/x3−1=elimx→∞xa1/x+b1/x+c1/x−33=elimx→∞xa1/x−1+b1/x−1+c1/x−13=elimx→∞13a1/x−11x+b1/x-11x+c1/x-11x=elimx→∞13loga+logb+logc=e13logabc=abc13

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$\underset{x \to \infty}{L i m} {(\frac{a^{1 / x} + b^{1 / x} + c^{1 / x}}{3})}^{x} =$

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${(a b c)}^{\frac{1}{3}}$

$- {(a b c)}^{\frac{1}{3}}$

answer is B.

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Detailed Solution

$\begin{matrix} \lim_{x \to \infty} {(\frac{a^{1 / x} + b^{1 / x} + c^{1 / x}}{3})}^{x} = e^{\lim_{x \to \infty} x (\frac{a^{1 / x} + b^{1 / x} + c^{1 / x}}{3} - 1)} \\ = e^{\lim_{x \to \infty} x (\frac{a^{1 / x} + b^{1 / x} + c^{1 / x} - 3}{3})} \\ = e^{\lim_{x \to \infty} x (\frac{a^{1 / x} - 1 + b^{1 / x} - 1 + c^{1 / x} - 1}{3})} \\ = e^{\lim_{x \to \infty} \frac{1}{3} (\frac{a^{1 / x} - 1}{\frac{1}{x}} + \frac{b^{1 / x} - 1}{\frac{1}{x}} + \frac{c^{1 / x} - 1}{\frac{1}{x}})} \\ = e^{\lim_{x \to \infty} \frac{1}{3} (\log a + \log b + \log c)} \\ = e^{\frac{1}{3} (\log a b c)} \\ = {(a b c)}^{\frac{1}{3}} \end{matrix}$