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  "body": "\n## Summary\n\nAsymptotic analysis describes the behaviour of functions near a limit point, such as $x \\to \\infty$ or a small parameter $\\varepsilon \\to 0$.\nIts basic tools are [[big O|0TGV80E6A4]] and [[little o|00T5EPFQB9]] notation, [[asymptotic equivalence|0QR92XBV6H]] and asymptotic expansions.\nAn asymptotic expansion need not converge, yet a few of its terms often give an excellent approximation.\nIn differential equations, it gives approximate solutions when exact ones are out of reach, as in perturbation methods.\n\n## Explanations\n\n### L1 human\n\nAsymptotic analysis asks what a function looks like when its argument becomes very large or very small.\nIt keeps the main terms and controls the size of what is left out.\nFor example, $x^2 + 3x + 7$ behaves like $x^2$ for large $x$, since the ratio tends to $1$.\nA finer statement is $x^2 + 3x + 7 = x^2 + 3x + O(1)$.\nStirling's formula $n! \\sim \\sqrt{2 \\pi n} \\, (n / e)^n$ is a famous result of this kind.\n",
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