Invariance of Euler-Lagrange equation

Personally I find the derivation slightly clearer when variations are written as $w(x) = \hat{w} (x) + \epsilon \beta(x)$ and the limits are performed with respect to the variation parameter $\epsilon$, not $\Delta \sigma$, for example as done in http://www.physics.usu.edu/Wheeler/ClassicalMechanics/CMCoordinateinvarianceofEulerLagrange.pdf.

Anyhow, I believe the point made by your textbook could be explained as follows.

If $$ \frac{ \Delta \sigma}{\Delta_1 \sigma } \to I $$ ($I$ standing for the Jacobian) as $\Delta \sigma, \Delta_1 \sigma \to 0$, one could write to first order $$ lim_{\Delta \sigma \to 0} \frac{J[y + h] – J[y]}{\Delta \sigma} = lim_{\Delta_1 \sigma \to 0} \frac{J[y + h] – J[y]}{I \Delta_1 \sigma} = \frac{1}{I} lim_{\Delta_1 \sigma \to 0} \frac{J[y + h] – J[y]}{\Delta_1 \sigma} $$ The action at the numerator is invariant upon coordinate changes, so the fact that $I \neq 0$ and that the first limit equals $0$ implies the conclusion.