<p>Based on paraxial and Wentzel–Kramers–Brillouin (WKB) approximations, we investigated density transitionbased self-focusing of asymmetric finite Airy-Gaussian (AiG) beams in plasma. The relativistic nonlinearity in the dielectric constant of plasma is considered. The electric field distribution of asymmetric finite AiG beams is expressed in terms of beam-width parameters<i> f</i>₁ and <i>f</i>₂ with respective modulation parameters <i>a</i> and <i>b</i> in transverse dimensions of the beam. The coupled differential equations governing the behavior of <i>f</i>₁ and <i>f</i>₂ with dimensionless propagation distance ξ are established from the parabolic equation approach. The effect of the exponential density ramp profile on the self-focusing of asymmetric finite AiG beams is presented. The sensitiveness of asymmetry in <i>a</i> and <i>b</i> parameters of finite AiG beam and exponential density ramp of plasma is specifically highlighted.</p>