# Lens combination

Suppose there are two lenses touching each other as shown in the diagram below. the focal length of the lenses are $f_1$ and $f_2$ respectively. Now by using the formula for lens combination I obtained the final focal length as $$\frac{f_1f_2}{f_1+f_2}$$.

Now suppose I assume there is a very small distance between them say d,where $d\ll R$, and I calculate the focal length of the system I still obtain the same result. Now my question is whether I can conclude from this that when ever two optical devices(lens,prism,etc) are touching each other. Can I do the calculation by assuming there is a small distance between them?

If there is a small gap between two optical components you have to do your calculations for a small separation. If the elements are touching each other then you have to set the separation $$d$$ to $$0$$.
If I understand your question correctly then the answer is yes, you can set the distance to $$0$$ before or after solving your system, due to continuity. Here, "before" means that you write your equations assuming that the components are touching each other, and "after" means that you account for a small gap, solve your equations and then set $$d=0$$ in the final result.
Click here to consider the case of 3 or more lenses, and try putting their separations to $$0$$.