= Solution
Choose an <orthonormal basis> $(e_1,e_2)$ of $T_pS$ and write $e(\theta)=\cos\theta\,e_1+\sin\theta\,e_2$. On a punctured disc on which the <exponential map> is a <diffeomorphism>, the map
$$
X(r,\theta)=\exp_p\bigl(r e(\theta)\bigr),
\qquad 0<r<\varepsilon,
$$
defines <geodesic polar coordinates> centred at $p$. The curves $r\mapsto X(r,\theta)$ are unit-speed <geodesics>, so the first coefficient of the <first fundamental form> is
$$
E=\langle X_r,X_r\rangle=1.
$$
The <Gauss lemma> says that radial and angular coordinate vectors are orthogonal, hence
$$
F=\langle X_r,X_\theta\rangle=0.
$$
The angular variation $J=X_\theta$ is a <Jacobi field> along each radial geodesic, with initial data
$$
J(0)=0,
\qquad
D_rJ(0)=e'(\theta),
\qquad |e'(\theta)|=1.
$$
Since $G=\langle X_\theta,X_\theta\rangle=|J|^2$, smooth dependence of the <geodesic flow> gives
$$
\sqrt{G(r,\theta)}=|J(r)|=r+o(r).
$$
Consequently
$$
\lim_{r\to0}G(r,\theta)=0,
\qquad
\lim_{r\to0}(\sqrt G)_r(r,\theta)=1,
$$
as required.
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