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  "body": "\n## Definition\n\nLet $n \\ge 2$, let $B = \\set{x \\in \\R^n : \\abs{x} \u003c 1}$ and let $S = \\partial B$.\nFor $\\zeta \\in S$ and $\\alpha \u003e 1$, the *nontangential approach region* at $\\zeta$ is\n$$\\Gamma_\\alpha(\\zeta) = \\set{x \\in B : \\abs{x - \\zeta} \u003c \\alpha (1 - \\abs{x})}.$$\nLet $u \\colon B \\to \\C$ and $L \\in \\C$.\nThe function $u$ has *nontangential limit* $L$ at $\\zeta$ if, for every $\\alpha \u003e 1$, $u(x) \\to L$ as $x \\to \\zeta$ with $x \\in \\Gamma_\\alpha(\\zeta)$.\n\n## Remarks\n\nThe limit along $\\Gamma_\\alpha(\\zeta)$ is a [[limit of a function of several variables|0J8Y1NCY0K]] restricted to that set; $\\zeta$ is a limit point of each $\\Gamma_\\alpha(\\zeta)$.\nFor the [[unit disc|8G5ABXRK85]] the regions $\\Gamma_\\alpha(\\zeta)$ are the Stolz angles, and the same definition works for the upper half-space with cones $\\set{(x, t) : \\abs{x - x_0} \u003c \\alpha t}$.\nA nontangential limit implies the radial limit $\\lim_{r \\to 1} u(r\\zeta) = L$.\nExample: let $f(z) = \\exp\\left(-\\frac{1 + z}{1 - z}\\right)$ on the unit disc and $\\zeta = 1$.\nSince $\\operatorname{Re} \\frac{1 + z}{1 - z} = \\frac{1 - \\abs{z}^2}{\\abs{1 - z}^2}$, on $\\Gamma_\\alpha(1)$ this is at least $\\frac{1}{\\alpha^2 (1 - \\abs{z})}$, which tends to $+\\infty$.\nSo $f$ has nontangential limit $0$ at $1$.\nOn the circle $\\abs{z - 1/2} = 1/2$ the same real part equals $1$, so $\\abs{f} = e^{-1}$ there, and $f$ has no unrestricted limit at $1$.\nNon-example: the Poisson kernel $u(z) = \\frac{1 - \\abs{z}^2}{\\abs{1 - z}^2}$ has $u(r) = \\frac{1 + r}{1 - r} \\to +\\infty$, so it has no finite nontangential limit at $1$.\nBy [[Fatou's theorem|Q2N917JQ6A]], the Poisson integral of an integrable function has a nontangential limit at almost every boundary point.\n",
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