Is there such a thing as $$\lim_{x\to\infty}\text{frac}(x)$$

What would be the reasoning in trying to prove its existence or non-existence?

  • $\begingroup$ Can you give us an example of what you mean by the fractional part of $x$? $\endgroup$ – DMcMor Jul 11 '17 at 15:07
  • $\begingroup$ By fractional part do you mean the decimal expansion of x? $\endgroup$ – Ollie Jul 11 '17 at 15:07
  • $\begingroup$ yes. like for 1,5 the fractional part is 0.5, for 20,56 its 0.56 $\endgroup$ – Alexandra Jul 11 '17 at 15:08
  • 1
    $\begingroup$ This limit does not exist. You can show that given any $L$, $N$, and $\epsilon$ there exists an $x>N$ such that $|L-\{x\}|>\epsilon$. This shows that $L$ cannot be the limit of the fractional part of $x$. $\endgroup$ – Callus Jul 11 '17 at 15:08
  • 2
    $\begingroup$ Consider the following increasing sequences for $x$: $1, 2, 3, 4, ...$ and $1.9, 2.9, 3.9, 4.9, ...$. What does $\text{frac}(x)$ converge to in both cases? $\endgroup$ – Tob Ernack Jul 11 '17 at 15:19

As noted in the comments, the limit does not exist. For clarification, here $\text{frac}(x)$ is the fractional part of $x$, which for positive $x$ is $\text{frac}(x) = x - \lfloor x \rfloor$. Plotted, this function looks like a 'sawtooth' and oscillates with period 1 and amplitude 1, so we would graphically not expect the limit to exist. There are two other ways to see this.

There are two increasing sequences that tend to infinity: $x_n = n$ and $y_n = n + 0.9$. For the first sequence $\lim_{n\rightarrow\infty} \text{frac}(x_n) = 0$ and for the second sequence $\lim_{n\rightarrow\infty} \text{frac}(y_n) = 0.9$, thus the limit $\lim_{x\rightarrow\infty} \text{frac}(x)$ does not exist.

The other way to see this is from the definition of this limit. For any $L$, there exists an $\epsilon$ such that for every $N$, there exists an $x>N$ such that $|L - \text{frac}(x)| > \epsilon$. (When $L \neq 0$, take $\epsilon = \frac{|L|}{2}$ and $x = \lceil N \rceil + 1$. When $L = 0$, take $\epsilon = \frac{1}{2}$ and $x = \lceil N \rceil + 0.9$.)


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.