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By starting from a generic metric which describes four dimensional
stationary black holes in an arbitrary theory of gravity , we show
that the $AdS$ near horizon geometry is a consequence of finiteness
and double-horizon limit. For the neutral and electrically charged
rotating black holes in some theories of gravity such as $f(R)$
gravity which are described by this general metric , we show that
applying this limit in the equations of motion results a set of
decoupled equations at the horizon which can be solved and give the
near horizon parameters. It is shown that these decoupled equations
come from variation of a function like entropy function which is
evaluated at the horizon by imposing double-horizon limit and
without going to the near-horizon coordinate. We simplify Wald
formula at the horizon by using this limit and get the results
similar to the entropy function method.
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