Given a differentiable function , then what is
$\lim_{x\to0} \left({ \frac{f(a)}{f(a+x)}}\right)^{\frac2x} $
Where $a$ is a real number.
If I use the identities $ \ln(1+x) \sim x $
and $ f(a+x) \sim f(a)+xf'(a) $
and take logarithms to both sides my guess is that the limit is
$ \exp\left(- 2\frac{f'(a)}{f(a)}\right) .$
Is this method correct with this result ? thanks in advance