  In this paper, we establish the precise asymptotic behaviors of the tail
probability and the transition density of a large class of isotropic L\'evy
processes when the scaling order is between 0 and 2 including 2. We also obtain
the precise asymptotic behaviors of the tail probability of subordinators when
the scaling order is between 0 and 1 including 1.
  The asymptotic expressions are given in terms of the radial part of
characteristic exponent $\psi$ and its derivative. In particular, when
$\psi(\lambda)-\frac{\lambda}{2}\psi'(\lambda)$ varies regularly, as
$\frac{t\psi(r^{-1})^2}{\psi(r^{-1})-(2r)^{-1}\psi'(r^{-1})} \to 0$ the tail
probability $\mathbb{P}(|X_t|\geq r)$ is asymptotically equal to a constant
times $ t( \psi(r^{-1})-(2r)^{-1}\psi'(r^{-1})).$
