HMMT 2008 K7

Find $p$ so that $\lim _{x \rightarrow \infty} x^{p}(\sqrt[3]{x+1}+\sqrt[3]{x-1}-2 \sqrt[3]{x})$ is some non-zero real number.

Dataset text NuminaMath olympiads. Quoted with source.

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\frac{5}{3}

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