Help me understand the question (existence of a limit in a function)

Question

I do not understand what the question is asking. The question goes like this:

Indicate those conditions on function $f: [0, +\infty) \to \mathbb{R}$, that guarantee the existence of the limit $\lim_{x \to + \infty} f(x)$. Do not forget, that $n$ is a natural number (everywhere).

And then there are a bunch of conditions. I will give three of them as an example.

• There exists a limit for $\lim_{n \to \infty} f(n)$
• For any $a \geq 0$, there exists a limit for $\lim_{n \to \infty} f(a+n)$
• For any $a > 0$, there exists a limit for $\lim_{n \to \infty} f(an)$.

I do not quite understand what the question is asking. If I interpret the question in a way that the arguments in those three options, namely $n, a+n, an$ are sequences. For example, $x_n = n, x_n=a+n, x_n=an$, then I can check by Gane's (I am not sure what it is called in English, but that is how it sounds in Russian. It is this definition:

for any sequence {$x_n$} $\in$ $E$ ($x_n \neq a$) such that $\lim_{n \to \infty} x_n = a \implies \lim_{n \to \infty} f(x_n) = A$

definition whether there is a limit.

Let's look at the first example and denote $x_n = n$. We know that $\lim_{n \to \infty} x_n = +\infty$ and thus there is a contradiction. Because by Gane's definition we are supposed to have a limit, therefore there is no limit of $\lim_{n \to \infty} f(n)$. But it is given in the exercise, that the function does have a limit...

So I am confused about what this exercise wants.

Answer 1

For each condition you must answer the question: "If this condition is satisfied then is it guaranteed that $\lim_{x\to\infty}f(x)$ exists?"

For example the answer concerning the first mentioned condition is: "no". This because we find an easy counterexample in the function that is prescribed by $x\mapsto\sin x$ if $x\notin\mathbb N$ and $x\mapsto 0$ otherwise.

Answer 2

The question is asking you which of the three conditions, if fulfilled, is enough for you to conclude that $\lim_{x\to\infty} f(x)$ exists.

In other words, you need to answer three sub-questions:

• Is it true that if there exists a limit $\lim_{n\to\infty} f(n)$ (in integers), then there exists a limit $\lim_{x\to\infty} f(x)$?
• Is it true that if, for every $a\geq 0$, there exists the limit $\lim_{n\to\infty} f(a+n)$, then there exists the limit $\lim_{x\to\infty} f(x)$?
• Is it true that if, for every $a>0$, there exists the limit $\lim_{n\to\infty} f(an)$, then there exists the limit $\lim_{x\to\infty} f(x)$?

You have to answer each of the three questions separately, either by proving that the statement is true, or by providing a counterexample.