Images Formed by Lenses

A converging lens has two regimes divided by the focal point: with the object outside $f$ the refracted rays cross and form a real inverted image, and with it inside $f$ they leave still diverging and form a virtual, upright, enlarged image, which is how a magnifying glass works. A diverging lens gives an upright reduced virtual image at every object distance. The thin-lens equation $1/s_i + 1/s_o = 1/f$ is a sum of reciprocals, so the algebra stays in reciprocal space until one final inversion. All three principal rays leave the tip of the object and change direction exactly once, at the line through the lens centre.

Three errors dominate. Assuming a converging lens gives a real inverted image wherever the object sits, so an object inside the focal length reads as a broken setup rather than as a magnifying glass. Solving the thin-lens equation as though it read $s_i + s_o = f$, or reaching $1/s_i$ and reporting that value as the distance. And tracing the principal rays from the base of the object rather than the tip, or bending them at the glass surfaces or at a focal point instead of at the lens plane.