Extending a geodesic

Question

I am working in prolate spheroidal coordinates $(\chi,\theta,\phi)$, defined by \begin{equation} \begin{split} x&=\sinh(\chi)\sin(\theta)\cos(\phi), \\ y&=\sinh(\chi)\sin(\theta)\sin(\phi), \\ z&=\cosh(\chi)\cos(\theta). \end{split} \end{equation} Given is a curve, parametrised as \begin{equation} \begin{split} x(\lambda)&=\cos(\lambda),\\ y(\lambda)&=\sin(\lambda), \\ z(\lambda)&=\sqrt{2\sinh(2\lambda)}, \end{split} \end{equation} and it is claimed to be a geodesic. This is well-defined for $\lambda\ge0$, and one can show that \begin{equation} \begin{split} \chi(\lambda)&=\ln(e^\lambda+\sqrt{e^{2\lambda}+1}), \\ \sin(\theta(\lambda))&=e^{-\lambda}, \\ \phi(\lambda)&=\lambda. \end{split} \end{equation} The question then asks to whether this geodesic (i.e. this curve) can be extended, i.e. whether we can make some continuation $x^\prime(\lambda),y^\prime(\lambda),z^\prime(\lambda)$ for $\lambda\le0$. My first thought was to take $x^\prime=x$ and $y^\prime=y$ and $z^\prime=-\sqrt(2\sinh(-2\lambda))$, but looking at some plots, $\tilde x^\prime=x(-\lambda), \tilde y^\prime=y(-\lambda)$ and $\tilde z^\prime=-\sqrt{2\sinh(-2\lambda)}$ also looked good. However, the tangent lines at zero do not coincide for $\lambda$ when you choose $\tilde x^\prime,\tilde y^\prime,\tilde z^\prime$, so this does not seem to be a valid option. But then I noticed something which disturbed me even more: the velocity at $\lambda=0$ is "$\infty$"? How can I understand this? Am I doing something wrong? The plots look smooth but the infinite velocity is disturbing, so what is going on here? How could you extend it looking at the curve in $(\chi,\theta,\phi)$ coordinates?