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$\lim\limits_{x\to\infty}f(x)^{1/x}$ where $f(x)=\sum\limits_{k=0}^{\infty}\cfrac{x^{a_k}}{a_k!}$.

Suppose $\{a_k\}$ be an strictly increasing sequence with positive integers.

Let $f(x)=\sum_{k=1}^\infty \cfrac{x^{a_k}}{a_k!}$. does $\lim\limits_{x\to +\infty}f(x)^\frac{1}{x}$ exist? and how to compute.

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Leitingok
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  • Since $f(x) \leq e^x$ for $x \geq 0$, $f(x)^{1/x}$ is bounded above by $e$. Thus both limsup and liminf exist. But I think that some deliberate choice of $(a_k)$ can make $f(x)^{1/x}$ oscillate as $x \to \infty$. – Sangchul Lee Nov 11 '12 at 12:00

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