Is $\infty / \infty = 1$? Lately, my friend and I were arguing about what $\infty / \infty$ equals.
My thinking was that $\infty / \infty = 1$, since no matter how high you go in the numerator, it would have to go equally as high in the denominator.
My friend pointed out that one is not the smallest it can go, and can be divided an infinite number of times. (Equaling $.\overline{0}$1)
Which is it? Or is infinity not even considered a real number and so the answer is really just undefined?

The following limits all have the indeterminate form of $\frac{\infty}{\infty}$, but they are not all $1$.
$$\lim \limits_{x \to \infty} \frac{x^2}{x}$$
$$\lim \limits_{x \to \infty} \frac{x}{x^2}$$
$$\lim \limits_{x \to \infty} \frac{x}{x}$$
However, if you are given $\frac{\infty}{\infty}$ without context, the value is indeterminate. Furthermore, note that $\infty$ is not a number, so it doesn't follow the standard rules of algebra.
We can take this one step further. $\lim \limits_{x \to \infty} \frac{x^2}{x}$ is infinite, and so is $\lim \limits_{x \to \infty} \frac{x^3}{x}$ -- their limits are the same. But doesn't that feel a bit strange? Wouldn't $x^3$ be "larger" because it's to the third power, not just the second? Well, now if we divide them, we get $\large{\lim \limits_{x \to \infty} \frac{\frac{x^3}{x}}{\frac{x^2}{x}}}$, which is $\infty$.
The conclusion overall being that, when comparing two infinite quantities, their relative growth rates -- "how fast they become infinite" -- must be considered.
