Lateral magnification–angular magnification relationship for a simple magnifier

In discussions of a simple magnifying lens, lateral magnification and angular magnification are usually treated as separate, unrelated entities, perhaps because the lateral magnification for a collimating magnifier goes to infinity, while the angular magnification remains finite. However, the angular magnification is equal to a ratio of the total lateral magnification for the combined magnifying lens–observation system to the initial lateral magnification for the observation system. The total lateral magnification for the combined magnifying lens–observation system remains finite for a collimating magnifier because the lateral magnification for the observation system goes to zero, while the lateral magnification for the magnifying lens goes to infinity. The resulting indeterminate form can be evaluated by limit techniques and is finite. Ways to avoid the indeterminate form are also discussed.