THIS IS NOT A PROPER ANSWER: this may or may not be about number theory, and that may or may not be a topic we cover on this calculator forum. but i like it so imma see what i can do

$${\frac{{\mathtt{\infty}}}{\left({\mathtt{\infty}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}} = {\frac{\infty}{{\mathtt{\,-\,}}{\mathtt{1}}{\mathtt{\,\small\textbf+\,}}\infty}}$$

thanks calculator.

lets see, an infinite value divided by one less than that infinite value.

well, as numerator value goes up (1/1, 2/1, 3/1) value goes up.

and as denominator goes down (1/3, 1/2, 1/1) value goes up also.

$${\frac{{\mathtt{\infty}}}{\left({\mathtt{\infty}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}}$$ would follow the same pattern as 5/4, 4/3, 3/2, 2/1, and 1/0 (which is interesting also because 1/0 is sometimes considered to be infinity)

by heading into the equation calculator and entering$${\mathtt{x}} = {\frac{{\mathtt{y}}}{\left({\mathtt{y}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}}$$ and$${\mathtt{y}} = {\mathtt{1}}$$

and then repeatedly adding 1s to y every time i see what the result is, i have concluded that the bigger y is, the closer y/(y-1 gets to being 1. for example, 2/1 = 2

but then 3/2 = 1.5,

4/3=1/333...

5/4=1.25

and eventually i just put 1+1+111111111111111 or something and got 1.00000000006

so yeah it just heads toward zero. so infinity/infinity-1 would be either 1 or 1.000000... and an infinite string of zeroes followed by a single one.

but what do i know