A student of mine asked me why $+\infty -\infty$ and $\frac{\infty}{\infty}$ are considered indeterminate forms.
I think that this is an only apparently stupid question. In other words how can I show, using simple arguments, that $+\infty -\infty$ isn't equal to zero and $\frac{\infty}{\infty}$ isn't equal to 1?
On
The problem is that $\infty$ is not a number that you can not operate as it doesn't belong to real numbers or to any mathematical set. Mathematically $\infty$ doesn't exist. When you think of infinity you think about a number that tends to infinity. Indeed infinity doesn't exist. However, we can deal with numbers that tend to infinity in the way that we study their behavior as they get greater. I will show an example. Take two functions: $$f(x) = x$$ $$g(x)= e^x$$ Both functions grow indefinitely if x grows as well as both have positive derivative throughout all its domain. When we ask what $f(x)/g(x)$ will be when $x\rightarrow \infty$ (when x gets bigger and bigger) you can work out the solution with a calculator using values of x and you would get that if you make x bigger then $f(x)/g(x)$ will be $0$. However this is a limit, you are not substituting a number by infinity, you are increasing its value.
So, the result of a function tending to infinity divided by another function tending to infinity will depend on the functions you are dividing. And very important, the functions aren't infinity, you are just measuring the result of the divisions as $x$ grows.
Take $a_n = 2n$ and $b_n=n$.
Obviously, $\lim_{n\to\infty} a_n = \infty$ and $\lim_{n\to\infty} b_n = \infty$. By moving the limits around naively, you see that $+ \infty - \infty = \lim_{n\to\infty} a_n - \lim_{n\to\infty} b_n = \lim_{n\to\infty} (a_n-b_n) = \infty \neq 0$.
And $\frac{\infty}{\infty} = \frac{\lim_{n\to\infty} a_n}{\lim_{n\to\infty} b_n}= 2 \neq 1$.
Note: This is not intended to be rigorous, as the question was more of a philosophical one.