Limit of Modulo Operation

From ProofWiki
Jump to navigation Jump to search

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

By Range of Modulo Operation for Positive Modulus and Range of Modulo Operation for Negative Modulus we have:

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

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.

$\blacksquare$