$\textbf{tl;dr:}$ Given that $r$ is a definining function for the boundary of a conformally compact manifold, how does one show that the sectional curvatures tend to $-1$ if $|\mathrm{d}r|^2_{\bar g} = 1$ on the boundary?
$\textbf{Definitions:}$ Let $(M, g)$ be a pseudo-Riemannian manifold. Suppose $M$ can be identified with the interior of a smooth compact manifold with boundary $\overline M$. Let $\Sigma$ denote $\partial M$ so that $\overline M = \Sigma \sqcup M$. We say that $r$ is a defining function for $\Sigma$ if $r: M \to \mathbb{R}$ is smooth, $\mathcal Z(r) = \Sigma$, and $\mathrm{d}r \neq 0$ on $\Sigma$. ($\mathcal Z(r)$ denotes the zero locus of $r$, that is, the set of points in $M$ where $r$ vanishes.)
We say that $(M, g)$ is conformally compact if there is a defining function $r$ for $\Sigma$ such that $$\bar g = r^2 g$$ on $M$, and $g$ is a metric on $\overline M$.
We say that $g$ is asymptotically hyperbolic if $|\mathrm{d}r|_{\bar g}^2 = 1$.
Question: My interpretation of asymptotically hyperbolic is that the sectional curvatures should tend to $-1$, so I set out to do some calculations. Does $|\mathrm{d}r|_{\bar g}^2 = 1$ really ensure that sectional curvatures tend to $-1$?
I start by using the conformal transformation rule for the Riemann curvature, and write $$\bar R_{ijkl} = r^2(R_{ijkl} + (\Upsilon_{ij} - \Upsilon_i \Upsilon_j + \frac{1}{2}\Upsilon^2 g_{ij})\circledast g_{kl})$$ where I have used $\circledast$ to denote the Kulkarni-Nomizu product. Note that $\Upsilon_{i} = r^{-1}\partial_{i}r$. I now proceed to write each term on the right as $r^kA_{ijkl}$ where $A_{ijkl}$ is bounded (in the sense that each component of the tensor is finite). First we note the following: \begin{align*} \Upsilon_{ij} &= \nabla_{i}\Upsilon_j = \nabla_i(r^{-1}\partial_j r) = r^{-1}\Phi_{ij} - r^{-2}\Phi_i \Phi_j\\ \Upsilon_i \Upsilon_j &= r^{-2}\Phi_i \Phi_j\\ \Upsilon^2 &= r^{-2}g^{ij}\Phi_i \Phi_j = r^{-2}r^{2}\bar g^{ij}\Phi_i\Phi_j = |\mathrm{d}r|^2_{\bar g} \end{align*} where $\Phi_i = \partial_i r$. Therefore, we have \begin{align*} R_{ijkl} &= r^{-2}\bar R_{ijkl} - (\Upsilon_{ij} - \Upsilon_i \Upsilon_j + \frac{1}{2}\Upsilon^2 g_{ij})\circledast g_{kl}\\ &= r^{-2}\bar R_{ijkl} - (r^{-1}\Phi_{ij} - r^{-2}\Phi_i \Phi_j - r^{-2}\Phi_i \Phi_j + \frac{1}{2}|\mathrm{d}r|^2_{\bar g} g_{ij})\circledast g_{kl}\\ &= r^{-2}\bar R_{ijkl} - (r^{-3}\Phi_{ij} - r^{-4}\Phi_i \Phi_j - r^{-4}\Phi_i \Phi_j + r^{-4}\frac{1}{2}|\mathrm{d}r|^2_{\bar g} \bar g_{ij})\circledast \bar g_{kl}. \end{align*} If the two terms in the "centre" of the expression ($r^{-4}\Phi_i \Phi_j$) were to cancel, it now becomes clear that the $|\mathrm{d}r|^2_{\bar g}$ term dominates as $r$ vanishes, and the expression directly implies that the sectional curvature tends to $-|\mathrm{d}r|^2_{\bar g}$. However, the signs are negative for both terms and the limits don't work out. Did I make a calculation mistake somewhere? Any pointers would be appreciated!
The problem with your computation is that $\Phi_{ij}$ is not bounded.
Your convention seems to be that $\Phi_{ij}$ represents the components of the second covariant derivative of $r$ with respect to the metric $g$. If you work in coordinates that are smooth up to the boundary, then the Christoffel symbols of $g$ will involve derivatives of $r^{-2}$ multiplied by components of $g^{ij} = r^2 \overline g^{ij}$, and therefore $\Phi_{ij}$ will be $O(r^{-1})$.
The powers of $r$ will become much easier to keep track of if you write $g = r^{-2} \overline g$ and compute $R_{ijkl}$ in terms of $\overline R_{ijkl}$, taking all of your covariant derivatives with respect to $\overline g$ instead of $g$.