Definition of coordinate basis vectors for manifolds

Question

I am currently learning about manifolds from Hobson's General Relativity: An Introduction for Physicists.

In Chapter 3 on 'Vector Calculus on Manifolds', pg. 57, the author defined coordinate basis vectors for a point $P$ on a $N$-dimensional manifold with coordinate system $x^a$ as $$\vec{e_a}=\lim_{\delta x^a \rightarrow 0}\frac{\delta \vec{s}}{\delta x^a}$$ where $\delta \vec{s}$ is the infinitesimal vector displacement between point $P$ and a nearby point $Q$ whose coordinate separation from $P$ is $\delta x^a$ along the $x^a$ coordinate curve.

Say if $P$ and $Q$ are points on the $x^1$ coordinate cuve, the coordinate separation $\delta x^1$ between them will be $0$ since on the coordinate curve the $x^1$ coordinate stays constant. If this is correct, the definition for $\vec{e}_a$ will not make sense since it is dividing by $0$. What am I getting wrong?

Answer 1

Maybe it becomes clear for the example of polar coordinates in the plane:

$$\mathbf e_r = \left(\frac{\partial x}{\partial r} , \frac{\partial y}{\partial r}\right) = (\cos(\theta) , \sin(\theta))$$

$$\mathbf e_{\theta} = \left(\frac{\partial x}{\partial \theta} , \frac{\partial y}{\partial \theta}\right) = (-r\sin(\theta) , r\cos(\theta))$$

The derivatives are the limits mentioned.