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.
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2Infinity is not a number; the notation simply means increasing without bound – J. W. Tanner Oct 28 '19 at 18:10
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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.
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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
$$f(x)=\sin(2x\pi), \quad f(n)=\sin(2n\pi)=0$$
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1@CameronWilliams Why not, if $n$ and $x$ would tend to $ \infty$ in the same way the limit should be the same. – user Oct 28 '19 at 18:10
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3OP asked if the "infinity" in both limits were the same, not about whether the limit themselves are equal. – 79037662 Oct 28 '19 at 18:17
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@79037662 Yes I get tour point, indeed I've added something more. Anyway I think it is important to point out that the two ways are not equivalent. – user Oct 28 '19 at 18:19