Lenses and Infinity: Exploring Image Formation at Infinity

[fisico30]

Hello, here is my question: if an object position can only tend to infinity, without ever reaching it (since infinity is just an abstraction), its image will tend to appear on the back focal plane of a converging lens, without ever forming there. The image will always be created on the image plane, infinitesimally far from the back focal plane.

If the object is instead placed exactly at the focal length (that is feasible), its image is said to be: upright, with finite magnification, and forming at infinity!
We all know that to view a virtual image we extrapolate the path of the rays.
If the image is at infinity the rays from each obj point are all parallel. The eye can be relaxed in viewing that virtual object. But we also know that if we look inside a simple magnifier (with object at f) the image appear to be some finite distance inside the lens, and not at infinity.
Where is the flaw in my thinking?

 

[HallsofIvy]

you do NOT look at an object at the focal point in a magnifier. That would cause the image to "blow up". You look at an object inside the focal distance.

 

[fisico30]

Ah, ok! Thank you!
In the book Physics by Giancoli it is said that if the obj distance is d_o=f the angular magnification M= N/f (N=25 cm for normal eye) and the image forms at infinity.
But I guess a better description of what happens is: if d_0 TENDS to f (the concept of limit), then the magnification M--> N/f. If d_0 was exactly equal to f, no image would be seen (as HallsofIvy points out). The distance f marks the transition between a real image and a virtual image formation. However, I am still wondering where this virtual image will actually appear to be: will the image distance tend to an undetermined, finite,large distance as d_0 -> f?

Giancoli also states that the angular magnification can be a little bit increased (+1) if the image forms at the near point N. In that case d_0 need to be less than f and exactly Nf/(N+f). In this situation the eye is not relaxed (ciliary muscles are working).

In both cases there is the strong assumption that the eye is almost touching the magnifying lens. This can be reasonable in the case of a microscope, less accurate in the case of spectacles.