Is infinity divided by infinity equal to infinity?

First off, infinity is not a number in conventional mathematics.

In Calculus, you can work with infinity through the language of limits. It is important to note that when we use the shorthand:

∞/∞

What we are REALLY saying is "the limit of a function which diverges to infinity divided by the limit of a function which diverges to infinity". We are not actually saying "infinity divided by infinity".

Now that THAT is out of the way, we can get to the answer.

∞/∞ is of indeterminate form, meaning that the division could converge to 0, it could converge to 1, it could converge to an arbitrary constant, or it could diverge to infinity.

In order to figure out which of these cases is true, you need to apply L'Hospital's rule, by taking the derivative of the numerator and the denominator (separately).