ansatz\]), we can employ the formulas discussed in the section \[section strip BH f(z)\].\
$\bullet$ [**Schwarzschild black hole.**]{} As a first example, we consider the Schwarzschild black hole, which is given by (\[charged BH metric z\]) with $Q=0$. By performing the expansion described above, we find $$L\,=\,
-\,\frac{\sqrt{2}\,z_0}{\sqrt{d(d+1)}}\,\log(z_0-z_{\textrm{max}})
+ O(1)$$ and $$\label{expansion nh shwarz}
\mathcal{A}_d(z_{\textrm{max}},0)
\,=\,
-\,\frac{\sqrt{2}}{\sqrt{d(d+1)}\,z_0^{d-1}}\,\log(z_0-z_{\textrm{max}})
+ O(1)
\,=\,
\frac{L}{z_0^{d}}
+ O(1)$$ where we recall that the horizon $z_0$ is related to the temperature as $T=(d+1)/(4\pi z_0)$. The case $d=3$ was considered in [@Ryu:2006ef].

$\bullet$ [**Extremal charged black hole.**]{} When $Q\neq 0$ and $T=0$ this analysis leads to $$L\,=\,
\frac{\sqrt{2}\,\pi\,z_0^{3/2}}{d \sqrt{d+1}\,\sqrt{z_0-z_{\textrm{max}}}}
+ O(1)$$ and $$\label{expansion nh rn T0}
\mathcal{A}_d(z_{\textrm{max}},0)
\,=\,
\frac{\sqrt{2}\,\pi\,z_0^{3/2-d}}{d \sqrt{d+1}\,\sqrt{z_0-z_{\textrm{max}}}}
+ O(1)
\,=\,
\frac{L}{z_0^{d}}
+ O(1)\;.$$ In the figure \[finite area L\] (see [@Barbon:2008sr]) we show $\mathcal{A}_d(z_{\textrm{max}},0)$ in terms of $L$ for the extremal case. When $L$ is small we are close to the boundary and the curve reproduces the one of $AdS_4$, as expected. By comparing the two plots in the figure, one can check the dependence on $z_0$ in (\[expansion nh rn T0\]).

  ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
   ![Extremal charged black hole in $AdS_4$ with $z_0=1$ (left) and $z_0=1.5$ (right). Plot of the finite term $\mathcal{A}_2(z_{\textrm{max}},0)$ as 