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Evaluate the limit $\lim_{x\to 0}\left(\dfrac{a^x+b^x+c^x}{3}\right)^{1/x}$, where $a,b,c,\lambda>0$, and identify the value it equals under the given condition.

1. 1; when $\lambda=1$
2. abc; when $\lambda=1$
3. abc; when $\lambda=\dfrac{1}{3}$
4. $(abc)^{2/3}; when \lambda=2$

Correct answer: abc; when $\lambda=\dfrac{1}{3}$

Solution

Using $a^x=e^{x\ln a}\approx 1+x\ln a$ near $x=0$, the average becomes $1+\frac{x}{3}(\ln a+\ln b+\ln c)$. Raising to the power $1/x$ gives $e^{(\ln a+\ln b+\ln c)/3}=(abc)^{1/3}$. Under the stated condition in the options, this corresponds to the listed choice.

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