What are sine and cosine doing here?

Calculus Level 5

\[\large \lim_{x \to \infty} \dfrac{f^{-1}(1000x)-f^{-1}(x)}{\sqrt [3]{x}} \]

Let \(f(x) = 27x^3 + (\cos13 + \sin13) x \), and let \(f^{-1}(x) \) denote the inverse function of \(f(x) \).
Compute the limit above.

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