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  "body": "\n## Definition\n\nThe hyperbolic tangent is the function $\\tanh \\colon \\mathbb{R} \\to \\mathbb{R}$ defined by\n$$\\tanh x = \\frac{\\sinh x}{\\cosh x} = \\frac{e^x - e^{-x}}{e^x + e^{-x}}.$$\n\n## Remarks\n\nThe quotient is defined everywhere, since $\\cosh x \\ge 1$.\nIt is odd and strictly increasing, with range $(-1, 1)$. Also $\\tanh x \\to \\pm 1$ as $x \\to \\pm \\infty$.\nMathlib defines it as `Real.tanh`.\n\n## Explanations\n\n### L1 human\n\nThe hyperbolic tangent is an S-shaped curve between $-1$ and $1$.\nIt is close to $x$ near $0$ and close to $\\pm 1$ far away.\n",
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