$\frac{\lfloor x \rfloor}{x} \to 1$ as $x \to \infty$?

Question

Could anyone be able to explain to me why $\frac{\lfloor x \rfloor}{x} \to 1$ as $x \to \infty$?

Answer 1

For every $x \ge 0$, we have $x-1 < \lfloor x \rfloor \le x$, thus $$ \frac{x-1}x < \frac{\lfloor x \rfloor}x \le \frac xx = 1. $$ By the sandwich theorem, you have your result.

Answer 2

Fill in details:

Putting $\;x-\lfloor x\rfloor=\{x\}=\;$ the fractional part of $\;x\;$ , one gets:

$$\frac{\lfloor x\rfloor}x=\frac{x-\{x\}}x=1-\frac{\{x\}}x\xrightarrow[x\to\infty]{}1$$

Answer 3

Because, if $x > 0$, $x = \lfloor x \rfloor +\{ x \} $ (integer part plus fractional part).

Therefore $\dfrac{\lfloor x \rfloor}{x}-1 =\dfrac{\lfloor x \rfloor-x}{x} =\dfrac{-\{ x \}}{x} \to 0 $ as $x \to \infty$ since $0 \le \{ x \} < 1$.

Answer 4

Think of $x$ as some amount of money which needs dollars and cents to express something like 355 dollars and 78 cents. If we are not to mention cents then we can say this amount is \$355.78

When the amount gets very high like millions and billions, its value is nearly as good if we omit the cents and mention only whole dollars. This is what the floor function does. What is the percentage loss when you are literally short-changed , say instead of getting a payment of \$300258 plus 93 cents you were given just \$300258.

Mathematical symbolism is a way of expressing precisely some statements. It helps understanding to go back to the less precise statement in words that it is being symbolized..