A small ball of mass m moves in a vertical circle at the end of a light inextensible string of length L. At the highest point the tension in the…

At the top, $T+mg = mv_t^2/L$ with $T=2mg$, so $v_t^2 = 3gL$. Moving from the top down to the horizontal position the ball falls through $L$, so by energy conservation $v_h^2 = v_t^2 + 2gL = 5gL$. At the horizontal position gravity is tangential, so the string tension alone supplies the centripetal force: $T = mv_h^2/L = 5mg$. The value $8mg$ is the tension at the lowest point, and $6mg$ would come from adding $mg$ wrongly at the horizontal position.