Could "$\infty$" be understood by taking the reciprocals of the Hyperreal numbers? When learning mathematics we are told that infinity is undefined. (*)
Recently I read about the infinitesimal version of Calculus and how we can in fact treat $dy/dx$ as a fraction under this approach (something we can't do with limits). This is achieved by constructing the Hyperreal numbers $^*\mathbb{R}$, which contain the real numbers and the infinitesimal numbers (all positive numbers greater than zero, yet less than any real number).
Thinking back to (*) I wonder, could sense of infinity be made somehow by including the reciprocal infinitesimal numbers? Presumably these numbers would be defined since the infinitesimal numbers are never zero.

No. The symbol $\infty$ as used in analysis usually means either positive infinite element of affinely extended real line (in this case you also would have $-\infty$ as another element) or (more rarely) the infinite element of the projectively extended real line.
In the later case it is the multiplicative inverse of zero ($\frac 10=\infty$).
So in fact you can use the symbol as part of a numerical system, but such system is called differently (affine real line or projective real line).
If you consider the hyperreal numbers, no hyperreal number has the properties usually ascribed to symbol $\infty$. For instance, for any hyperreal number $a$, $a+1\ne a$, while $\infty$ is usually defined as having the property of invariancy against addition: $\infty+1=\infty$. 
That said, you of course can extend the hyperreal numbers in affine or projective manner just as we do with real numbers by adding $\infty$ as an element but this is not different from how reals can be extended.
