  With the successes of $f(R)$ theory as a neutral modification of Einstein's
general relativity (GR), we continue our study in this field and attempt to
find general %natural { neutral} and charged black hole (BH) solutions. In the
previous papers \cite{Nashed:2020mnp,Nashed:2020tbp}, we applied the field
equation of the $f(R)$ gravity to a spherically symmetric space-time
$ds^2=-U(r)dt^2+\frac{dr^2}{V(r)}+r^2 \left( d\theta^2+\sin^2\theta d\phi^2
\right)$ with unequal metric potentials $U(r)$ and $V(r)$ and with/without
electric charge. {Then we have obtained equations which include all the
possible static solutions with spherical symmetry.} To ensure the closed form
of system of the resulting differential equations in order to obtain specific
solutions, we assumed the derivative of the $f(R)$ with respect to the scalar
curvature $R$ to have a form %$F_1(r)=\frac{df(R(r))}{dR(r)} \propto
%{\color{red} 1 +} %\frac{c}{r^n}$ but in case $n>2$, the resulting black hole
solutions with/without charge do not %generate asymptotically GR BH solutions
in the limit $c\rightarrow 0$ which means that the only case that can generate
GR BHs is $n=2$. %In this paper, we assume another form, i.e., $F_1(r)
{=\frac{df(R(r))}{dR(r)} } = 1-\frac{F_0-\left(n-3\right)}{r^n}$ with a
constant $F_0$ and show that we can generate asymptotically GR BH solutions for
$n>2$ but we show that the $n=2$ case is not allowed. This form of $F_1(r)$
could be the most acceptable physical form that we can generate from it
physical metric potentials that can have a well-known asymptotic form and we
obtain the metric of the Einstein general relativity in the limit of $F_0\to
n-3$. We show that the form of the electric charge depends on $n$ and that
$n\neq 2$.}
