Lately I have been having a discussion on infinity. Personally I do not see this as a number, but as a concept. The person I was speaking with, asked me to assume for the time being that $\infty$ is a number, however. Let us assume this for this question.
As we know, we have $\lim_{x \rightarrow \infty} \frac{1}{x} = 0$. I can understand this, as it is visible in a plot of the function $\frac{1}{x}$ that $x$ approaches zero if $x$ becomes very large.
However, now that we assume that $\infty$ is a number, can we conclude that $\frac{1}{\infty}=0$?
My friend thinks this makes sense, but I cannot wrap around this. Maybe my thinking is not developed enough to actually think of $\infty$ as a number.
Also, I am aware that there are different sizes of infinity, put simply. Does it matter for this question which one you think about?
The problem is the consequences of the hypothetical
Assuming "$\infty$ is a number" is implicitly assuming that it along with the ordinary numbers follows the usual rules of arithmetic. Then you find many contradictions. Once it's a number you have to allow $\infty + 1$. If that turns out to be $\infty$ then $1=0$. If it turns out to be some ordinary number then subtraction tells you $\infty$ is an ordinary number too.
There are many places in mathematics where it's convenient to use the symbol $\infty$. None of those treats it as a number.