Let $(X,d)$ be a strictly intrinsic metric space, that is, any pair of points in $X$ can be joined by a (not necessarily unique) rectifiable curve with the length being equal to the distance between these points. All curves further are assumed to be rectifiable and arc-length parametrized. Say that a curve $\gamma$ in $X$ is geodesic if it is locally length minimizing.
My question is as follows. Let $\gamma_0$ be a curve in $X$ with $\gamma_0(0)=p$. Suppose that $\gamma$ is "as good as possible" at $t_0=0$. For instance, suppose that \begin{equation} (*)\quad \lim\limits_{t\to +0} \frac{d(p,\gamma_0(t))}{t}=1. \end{equation} In general, one can require from the behavior of $\gamma$ at $t_0=0$ any property that holds for a.e. $t\in [0, \mathbf{l}(\gamma_0)]$ for a general curve, where $\mathbf{l}$ denotes the length of the corresponding curve. Can I find a sequence $(\gamma_n)_{n\in \mathbb{N}}$ of geodesic curves in $X$ with $\gamma_n(0)=p$ such that \begin{equation} \varlimsup\limits_{n\to +\infty} \varlimsup\limits_{t\to +0} \frac{d(\gamma_0(t),\gamma_n(t))}{t}=0? \end{equation} In other words, can I find a geodesic curve that is almost tangent to a given curve with the error being arbitrarily small? Under what conditions on $X$, or $p$, I can always guarantee that?
Some of my thoughts are below.
Condition $(*)$ means that, near $t_0=0$, the curve $\gamma_0$ is almost isometry, so probably this can be used. For examle, given $\varepsilon>0$, let $t_{\varepsilon}>0$ be such that \begin{equation} \frac{d(p,\gamma_0(t))}{t}\in [1-\varepsilon,1+\varepsilon] \end{equation} for any $t\in (0,t_{\varepsilon})$. Let $\gamma_{\varepsilon}$ be a length minimizing curve joining $p$ and $\gamma_0(t_{\varepsilon})$. It seems at first that this $\gamma_{\varepsilon}$ is a good candidate for approximating curve, but, as far as I understand, we can say almost nothing about how close $\gamma_0$ and $\gamma_{\varepsilon}$ to each other near $t_0=0$ for a general space. So, I guess I need some limitations on $X$.
I have a feeling that my question is connected with the metric geometry and the notions of angle, space of directions, or something like this. More precisely, I think that my statement is close to the following informal one: any sufficiently good curve is equivalent to a geodesic curve in the sence of the space of directions at $p$. And probably, for my statement to hold, one needs some curvature bounds on $X$. I am not deeply familiar with the metric geometry, so do not know whether my question has a standard answer.
Would be gratedul for any help!