  This paper is devoted to studying the Cauchy problem for the generalized
Ostrovsky equation \begin{eqnarray*}
u_{t}-\beta\partial_{x}^{3}u-\gamma\partial_{x}^{-1}u+\frac{1}{k+1}(u^{k+1})_{x}=0,k\geq5
\end{eqnarray*} with $\beta\gamma<0,\gamma>0$. Firstly, we prove that the
Cauchy problem for the generalized Ostrovsky equation is locally well-posed in
$H^{s}(\mathbb{R})\left(s>\frac{1}{2}-\frac{2}{k}\right)$. Then, we prove that
the Cauchy problem for the generalized Ostrovsky equation is locally well-posed
in $X_{s}(\mathbb{R}):
=\|f\|_{H^{s}}+\left\|\mathscr{F}_{x}^{-1}\left(\frac{\mathscr{F}_{x}
f(\xi)}{\xi}\right)\right\|_{H^{s}}\left(s>\frac{1}{2}-\frac{2}{k}\right).$
Finally, we show that the solution to the Cauchy problem for generalized
Ostrovsky equation converges to the solution to the generalized KdV equation as
the rotation parameter $\gamma$ tends to zero for data belonging to
$X_{s}(\mathbb{R})(s>\frac{3}{2})$. The main difficulty is that the phase
function of Ostrosvky equation with negative dispersive
$\beta\xi^{3}+\frac{\gamma}{\xi}$ possesses the zero singular point.
