Limit of Modulo Operation

Theorem
Let $x$ and $y$ be real numbers.

Let $x \bmod y$ denote the modulo operation.

Then holding $x$ fixed gives:
 * $\displaystyle \lim_{y \mathop \to 0} x \bmod y = 0$


 * $\displaystyle \lim_{y \mathop \to \infty} x \bmod y = x$ if $x \ge 0$