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We consider the Li\'{e}nard system in the plane and present general assumptions to obtain some new explicit conditions under which this system has or fails to have a positive orbit which starts at a point on the vertical isocline and approaches the origin without intersecting the $x$-axis. This arises naturally in the existence of homoclinic orbits and oscillatory solutions. Our investigation is based on the notion of orthogonal trajectories of orbits of the system.

Conditions for approaching the origin without intersecting the x-axis in the Liénard plane