# 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 $x \ge 0$

## Proof

### Limit 1

$- \size y \lt x \bmod y \lt \size y$

The result follows from the Squeeze Theorem.

$\blacksquare$

### Limit 2

As $y \to \infty$:

 $\displaystyle 0$ $\le$ $\, \displaystyle x \,$ $\, \displaystyle <\,$ $\displaystyle y$ $\displaystyle \leadsto \ \$ $\displaystyle 0$ $\le$ $\, \displaystyle \frac x y \,$ $\, \displaystyle <\,$ $\displaystyle 1$ $\displaystyle \leadsto \ \$ $\displaystyle$  $\, \displaystyle \floor {\frac x y} \,$ $\, \displaystyle =\,$ $\displaystyle 0$

Therefore by the definition of modulo operation:

 $\displaystyle \lim_{y \mathop \to \infty} x \bmod y$ $=$ $\displaystyle \lim_{y \mathop \to \infty} x - y \floor {\dfrac x y}$ $\displaystyle$ $=$ $\displaystyle \lim_{y \mathop \to \infty} x - y \cdot 0$ $\displaystyle$ $=$ $\displaystyle x$

Hence the result.

$\blacksquare$