  In this paper, the regularity results for the integro-differential operators
of the fractional Laplacian type by Caffarelli and Silvestre \cite{CS1} are
extended to those for the integro-differential operators associated with
symmetric, regularly varying kernels at zero. In particular, we obtain the
uniform Harnack inequality and H\"older estimate of viscosity solutions to the
nonlinear integro-differential equations associated with the kernels
$K_{\sigma, \beta}$ satisfying
  $$ K_{\sigma,\beta}(y)\asymp \frac{ 2-\sigma}{|y|^{n+\sigma}}\left(
\log\frac{2}{|y|^2}\right)^{\beta(2-\sigma)}\quad \mbox{near zero} $$ with
respect to $\sigma\in(0,2)$ close to $2$ (for a given $\beta\in\mathbb R$),
where the regularity estimates do not blow up
  as the order $ \sigma\in(0,2)$ tends to $2.$
