‎Find some examples which have this property $‎\lim‎_{n\to{‎\infty‎}}f(n+x)\neq\lim‎_{n\to{‎\infty‎}}f(n)$‎.

Question

Let ‎‎$‎‎f$ be a‎ ‎real ‎function. Can anyone help me giving an ‎example with the following property

‎‎‎$‎‎‎\lim‎_{n\to{‎\infty‎}}f(n+x)\neq\lim‎_{n\to{‎\infty‎}}f(n) \quad; n\in\mathbb{N}$‎

for this purpose, I think that I should follow the functions with the following property‎ ‎$$‎‎f(a+b)=f(a)f(b) $$‎‎‎‎‎ It is same as exponential function but it doesn't work.

Answer 1

Try $$f(x) = \sin(2\pi x).$$ Then $f(n) = 0$ for all $n$, so $\lim_{n \to \infty} f(n) = 0$. But, if we take $x = \frac{1}{4}$, then $$\lim_{n \to \infty} f(x + n) = \lim_{n \to \infty} 1 = 1.$$

Answer 2

Alt. hint:   let $\,f\,$ be the Dirichlet function and $\,x = \sqrt{2}\,$, then the limits are $\,0\,$ and $\,1\,$, respectively.