I am glad you asked and yes infinity over infinity is not always one............At least as we approach infinity. This is the start of the wonderful world of calculus.
Lets look at a couple of examples:
So you are correct in that one side can approach infinity faster than the other. We call these limits which is nothing more than saying we are getting incrementally close to infinity, zero, or another defined number.
Lets first look at something obvious:
x^2/x------------->infinity/infinity=1 as x approaches infinity right?
No, it approaches infinity.
Conversely x/x^2 approaches zero as x approaches infinity instead of 1.
Now, I know you probably saying hey wait a minute x^2/x=x and x/x^2=1/x. So one is infinity and the other is 1/infinity. Actually the way it is written both would be infinity/infinity but you defined something very important and that is a rate at which the numerator and the denominator approaches infinity. That is the fundamental principle here.
This actually will get quite a bit more advanced and there are many more principles to learn. This is approaching the definition of the fundamental principle of calculus. So to be concise your thesis is that one side will approach infinity faster and that is correct.
I hope this helps.