Is there a calculation in which infinities cancels out?

There are many asymptotic functions that are expressed as the x or y value approaches infinity. They are not at all uncommon.

And the ratio of two such functions would define something like a line. Students of Analytic Geometry as well as Calculus are quite familiar with the occurrence of infinity.

No. Infinity isn’t a number, so you can’t apply algebraic operations to it.

∞ – ∞ doesn’t make any sense, and neither does ∞/ ∞. In calculus, one might write something like this as a shortcut to evaluating a limit, but it’s important to keep in mind that we mean the limit as a variable approaches some number (or infinity).

There are some number systems that have infinity as part of the number line. I don’t know the details, but here are the links:
https://en.wikipedia.org/wiki/Projectively_extended_real_line
https://en.wikipedia.org/wiki/Extended_real_number_line

It depends on what you mean by a calculation. There is a simple proof using infinities that there are numbers that are not the solution of any finite polynomial with integer coefficients. The idea of the proof is that the collection of such polynomials is countable, meaning that they are as large as the infinity of whole numbers. The infinity of all real numbers is larger than that of the whole numbers. Therefore there must be some that are not represented by a polynomial. Such numbers are called transcendental. Although there are many more transcendental numbers than algebraic numbers, finding them is very difficult. Pi and e are among the few numbers known to be transcendental.

Physicists have to cancel out infinities in a process called renormalisation to get quantum theory to work.