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Doing some calculus papers before going university and I found this question:

Find $\lim_{x\to\infty} \left[\frac{1}{3} \left(3^\frac{1}{x} + 8^\frac{1}{x} + 9^\frac{1}{x} \right)\right]^x$

My solution was to use AM-GM, as shown: Since $3^\frac{1}{x} + 8^\frac{1}{x} + 9^\frac{1}{x} \geq 3 \sqrt[3]{3^\frac{1}{x} 8^\frac{1}{x} 9^\frac{1}{x}} = 3 \cdot 6^\frac{1}{x}$, it follows that $\lim_{x\to\infty} \left[\frac{1}{3} \left(3^\frac{1}{x} + 8^\frac{1}{x} + 9^\frac{1}{x} \right)\right]^x \geq \lim_{x\to\infty} \left[\frac{1}{3}\cdot 3 \cdot 6^\frac{1}{x} \right]^x=6$. Equality holds as $\lim_{x \to\infty} 3^\frac{1}{x} = \lim_{x \to\infty} 8^\frac{1}{x} = \lim_{x \to\infty} 9^\frac{1}{x}$

Is this solution rigorous enough? Or what would be a better way (if any) of solving this?

Thanks

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    $\begingroup$ Your statement 'Equality holds as $\lim_{x \to\infty} 3^\frac{1}{x} = \lim_{x \to\infty} 8^\frac{1}{x} = \lim_{x \to\infty} 9^\frac{1}{x}$' does not make sense. $\endgroup$
    – Kavi Rama Murthy
    May 27, 2022 at 9:03
  • $\begingroup$ @KaviRamaMurthy my apologies but could I understand why not? all of those 3 limits tend to 1 at infinity $\endgroup$
    – Russell Ng
    May 27, 2022 at 9:17
  • $\begingroup$ I think what OP wants to say is that AM=GM since the limits of all those terms considered in the inequality are equal. $\endgroup$
    – Doobius
    May 27, 2022 at 9:19
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    $\begingroup$ Alternatively write the limit from $(f(x))^x$ as $e^{x\ln(f(x))}$ then sub $y=\frac{1}{x}$ and apply L'Hôpital's rule $\endgroup$
    – Doobius
    May 27, 2022 at 9:23
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    $\begingroup$ @Doobius But this cannot be used here (at least not without further justification): For any given $x$, we only know the inequality, and raising the inequality to the power of $x$ might destroy equality for $x \to \infty$. $\endgroup$
    – LurchiDerLurch
    May 27, 2022 at 9:24

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