Equal Modulusly?

(For now, let's work in positive integers)

Even though the statement seems true, esp for small numbers, it is actually false. As a counter example, take $x = 1, y = 2, n = 2$ . We have

$x \times ( y \mod n) = 1 \times 0 = 0, \\ y \times ( x \mod n ) = 2 \times 1 = 2.$

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The true statement is

$\left( x \times ( y \mod n) \mod n \right) = \left( y \times ( x \mod n ) \mod n \right)$

where both values are equal to $(xy \mod n )$ . (Can you show this?) When we tried small values, most things happen to cancel out. However, in general, without taking the final mod n, we could have different values of the residue class.

Note: The original statement is true if $x , y < n$ , which is why "small cases work". To get a counter example, we will need one or $x \geq n, y \geq n$ .

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(Now, let's extend it to real numbers.)

The "true" statement is no longer true when applied to the reals. As a counter example, take $x = 1.1, y = 3.1, n = 1$ . Then,

$\left( x \times ( y \mod n) \mod n \right) = \left( 1.1 \times .1 \mod 1 \right) = 0.11 , \\ \left( y \times ( x \mod n ) \mod n \right) = \left( 3.1 \times .1 \mod 1 \right ) = 0.31 , \\ \left( xy \mod n \right) = \left( 3.41 \mod 1 \right) = 0.41$

Note: The original statement is true if $x , y < n$ . Again, to get a counter-example, we will need one or $x \geq n, y \geq n$ .

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Now, let's apply what we've learnt to complex exponentiation with evaluation at the principle branch of the argument: $[ 0, 2 \pi ]$ .
When we want to say that

$\left ( e ^ x \right) ^ y = e^{xy} = \left( e^y \right) ^ x,$

what we actually mean is

$\left( y \times ( x \mod 2 \pi ) \mod 2 \pi \right) = \left ( xy \mod 2 \pi \right) = \left( x \times ( y \mod 2 \pi ) \mod 2 \pi \right)$

However, because this statement is not always true, hence complex exponentiation with evaluation is not always true.

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In particular, is it true that

$\lim_{ x \rightarrow 0 } \left ( e^ x \right) ^ { \frac{1}{x} } = 1 ?$ $\lim_{ x \rightarrow 0 } \left ( e^ \frac{1}{x} \right) ^ { x} = 1 ?$

Hint: Try various subsequences like $x_n = \frac{1}{n}, \frac{ 1 } { 2n \pi }, \frac{ 2 \pi } { n}, \ldots$