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  "body": "\n## Statement\n\nFor every real $a \u003e 0$,\n$$\\lim_{x \\to \\infty} \\frac{\\ln x}{x^a} = 0.$$\nThat is, $\\ln x = o(x^a)$ as $x \\to \\infty$.\n\n## Remarks\n\nSubstitute $x = e^{t/a}$. Then $\\frac{\\ln x}{x^a} = \\frac{t / a}{e^{t}}$, which tends to $0$ by [[exponential growth|0CT6CBGTYY]].\nIn [[little o notation|04TSHB9H8D]], $\\ln x = o(x)$. Mathlib states this case as `Real.isLittleO_log_id_atTop`.\nSimilarly, $x^a \\ln x \\to 0$ as $x \\to 0^+$.\n\n## Explanations\n\n### L1 human\n\nThe logarithm goes to infinity, but very slowly.\nEach positive power of $x$, even $x^{0.001}$, eventually beats it.\n",
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