Limit of Modulo Operation

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Theorem

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

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


Then holding $x$ fixed gives:

$\ds \lim_{y \mathop \to 0} x \bmod y = 0$
$\ds \lim_{y \mathop \to \infty} x \bmod y = x$ 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:

$-\size y < x \bmod y < \size y$

The result follows from the Squeeze Theorem for Functions.

$\blacksquare$


Limit 2

As $y \to \infty$:

\(\ds 0\) \(\le\) \(\, \ds x \, \) \(\, \ds < \, \) \(\ds y\)
\(\ds \leadsto \ \ \) \(\ds 0\) \(\le\) \(\, \ds \frac x y \, \) \(\, \ds < \, \) \(\ds 1\)
\(\ds \leadsto \ \ \) \(\ds \) \(\) \(\, \ds \floor {\frac x y} \, \) \(\, \ds = \, \) \(\ds 0\)


Therefore by the definition of modulo operation:

\(\ds \lim_{y \mathop \to \infty} x \bmod y\) \(=\) \(\ds \lim_{y \mathop \to \infty} x - y \floor {\dfrac x y}\)
\(\ds \) \(=\) \(\ds \lim_{y \mathop \to \infty} x - y \cdot 0\)
\(\ds \) \(=\) \(\ds x\)

Hence the result.

$\blacksquare$