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Let $\Delta\subsetneq \mathbb C$ be a simply connected domain, let $f:\mathbbD \to \Delta$ be a Riemann map and let $\{z_k\}\subset \Delta$ be a compactlydivergent sequence. Using Gromov's hyperbolicity theory, we show that$\{f^{-1}(z_k)\}$ converges non-tangentially to a point of $\partial \mathbb D$if and only if there exists a simply connected domain $U\subsetneq \mathbb C$such that $\Delta\subset U$ and $\Delta$ contains a tubular hyperbolicneighborhood of a geodesic of $U$ and $\{z_k\}$ is eventually contained in asmaller tubular hyperbolic neighborhood of the same geodesic. As a consequencewe show that if $(\phi_t)$ is a non-elliptic semigroup of holomorphic self-mapsof $\mathbb D$ with K\"onigs function $h$ and $h(\mathbb D)$ contains avertical Euclidean sector, then $\phi_t(z)$ converges to the Denjoy-Wolff pointnon-tangentially for every $z\in \mathbb D$ as $t\to +\infty$. Using newlocalization results for the hyperbolic distance, we also construct an exampleof a parabolic semigroup which converges non-tangentially to the Denjoy-Wolffpoint but oscillating, in the sense that the slope of the trajectories is not asingle point.

Non-tangential limits and the slope of trajectories of holomorphic semigroups of the unit disc