$n \rightarrow \infty$ and $x\rightarrow \infty$

Question

If i have $\lim_{n \rightarrow \infty}s_{n}$ for all $n\in \mathbb{N}$ like a sequence and $\lim_{x\rightarrow \infty}f(x)$ for all $x\in \mathbb{R}$, Do $x$ and $n$ tend to the same infinity? i do not know if my question is well asked, i think the answer is yes, both are very large numbers on the same line, some idea or explanation thank you very much in advance.

Answer 1

Yes, if we consider the definition of limit we can say that $x$ and $n$ tend to $\infty$ according to the same definition since for any $M$ such that

• $\exists x_0\quad\forall x\ge x_0 \implies x\ge M$

• $\exists n_0 \quad\forall n\ge n_0 \implies n\ge M$

Anyway, pay attention to the fact that the ways $x$ and $n$ tend to $\infty$ are not equivalent, indeed let consider for example

$$f(x)=\sin(2x\pi), \quad f(n)=\sin(2n\pi)=0$$

Answer 2

If you consider $\mathbb N$ as a subset of $\mathbb R$, then yeah, kind of. For any $n\in \mathbb N$, there is a $x\in\mathbb R$ such that $x>n$, and for all $x\in\mathbb R$ there is a $n\in\mathbb N$ such that $n>x$.

So if you were to adjoin an element called $\infty$ to the real line such that $\infty=\sup\mathbb R$ (as we often do), then that same $\infty$ would be the supremum of $\mathbb N$ as well.