I know that if \limsup_{x\to\infty}f(x)
Is the following true:
If \limsup_{x\to\infty}f(x)
(If the above is true, does $x_{o}$ depends on $\epsilon$?) Thanks
I know that if \limsup_{x\to\infty}f(x)
Is the following true:
If \limsup_{x\to\infty}f(x)
(If the above is true, does $x_{o}$ depends on $\epsilon$?) Thanks
Yes, it is true and $x_0$ is independent of $\epsilon$ - just pick the same $x_0$ as you have in the first statement you said you know is true. Your second statement is a weaker version of your first.
Based on this comment is seems that OP had a slightly different question in mind. (But he did not edited the question. And he did not answered, what definition of the limit superior he is using.) Basically, without knowing what is his definition of $\limsup$ we should be working with, the question cannot be answered. Anyway, I'll mention here several possibilities how $\limsup$ can be defined. I'll also discuss (very briefly) why these definitions are equivalent. Hopefully, this will be helpful for the OP.
Definition 1. Let $S\in\mathbb R$. We say that $S=\limsup\limits_{x\to\infty} f(x)$ if the following two conditions are fulfilled:
For each $\varepsilon>0$ there exists $x_0$ such that $x>x_0 \Rightarrow f(x). $(\forall\varepsilon>0)(\exists x_0)(x>x_0 \Rightarrow f(x)
For each $\varepsilon>0$ and $M$ there exists $x>M$ such that $f(x)>S-\varepsilon$. $(\forall\varepsilon>0)(\forall M)(\exists x>M) f(x)>S-\varepsilon.$
Of course, limit superior can also be equal to $\pm\infty$. Definition has to be appropriately modified for these cases; I'll omit this for the sake of simplicity. (I have also ignored the question why such $S$ exists.)
Definition 2. $S=\sup\{L; \text{ there exists a sequence }(x_n)\text{ such that }x_n\to\infty, f(x_n)\to L\}$
Definition 3. $S=\lim\limits_{N\to\infty} \sup\{f(x); x\ge N\}$
All these definition can be modified to get $\limsup_{x\to x_0} f(x)$ instead of $\limsup_{x\to\infty} f(x)$.
Let us show that these definitions are equivalent. I.e., suppose that $S_1$, $S_2$, $S_3$ are the values given by the above definitions. We will assume that $f(x)$ is bounded (for the sake of simplicity), which implies that $S_{1,2,3}$ are finite.
It is easy to see that $\boxed{S_2\le S_3}$, since if $x_n\to\infty$ then $x_n>N$ for large $n$'s and $f(x_n) \le \sup\{f(x); x\ge N\}$.
Using definition 1 we can inductively construct sequence $x_n$ such that $x_n\to\infty$ and $|f(x_n)-S_1|<\frac1n$ (by choosing $\varepsilon=\frac1n$); which implies that $x_n\to S_1$. This implies $\boxed{S_1\le S_2}$.
If $N$ is large enough then, from definition 1, we have $\sup\{f(x); x\ge N\}\le S_1+\varepsilon$. This implies that $S_3\le S_1+\varepsilon$. Since this is true for any $\varepsilon>0$, we get $\boxed{S_3\le S_1}$.
Yes, it is true trivially since L
A stronger statement is (trivially) true also: there exists an $\epsilon>0$ and an $x_0$ such that f(x)