Lenses ``within'' Infinite Telescopes

We now illustrate by way of example that asymptotic curves look identical through an optical telescope.

The curves $f(x)=1/x$ and $g(x)=0$ are asymptotic. Let $H$ be a positive infinite hyperinteger number and point a $1$-lens in $(H,1/H)$. We can see that

$$\left(H+1,\frac{1}{H+1}\right) \mapsto \left(1,\frac{1}{H+1}-\frac{1}{H} \right) = \left(1,\frac{-1}{(H+1)H}\right)$$

and by taking the standard parts, the optical $1$-lens maps

$$\left(H+1,\frac{1}{H+1}\right) \mapsto (1,0)$$

as $-1/(H+1)H$ is infinitesimal. As such, a point in the form $(H+r,1/(H+r))$, where $r\in [0,1]$, is mapped in the point $(r,0)$, and this means that $f(x)$ is indistinguishable from $g(x)$ when $x$ is infinite.

However, we can point an astigmatic lens ``within'' the telescope in $(H,1/H)$, in order to visualize what really happens at infinity:

$$(x,y) \mapsto \left(x-H,\frac{y-\frac{1}{H}}{\frac{1}{H}}\right)$$

and, then, we obtain that a point on the hyperreal graph of $f$ is mapped again as follows

$$\left(H+r,\frac{1}{H+r}\right) \mapsto \left(r,\frac{\frac{1}{H+r}-\frac{1}{H}}{\frac{1}{H}}\right)= \left(r,\frac{-r}{H+r}\right) \mapsto (r,0)$$

and the corresponding point of $g$

$$(H+r,0) \mapsto (r,-1).$$

Figure 4: The graph of $f(x)=1/x$ through an optical telescope.

Hence, if we point a more powerful astigmatic lens within the telescope, we can see that the graphs of $f$ and $g$ are straight, distinct, and parallel lines at infinity.