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The formal definition of limit at infinity usually starts with a statement requiring an open interval. An example from OSU is as follows:

Limit At [Negative] Infinity: Let $f$ be a function defined on some open interval $(a, \infty)$ [$(-\infty, a)$]. Then we say the limit of $f(x)$ as $x$ approaches [negative] infinity is $L$, and we write $\lim_{x\to[-]\infty} f(x) = L$ if for every number $\epsilon$, there is a corresponding number $N$ such that $\left|f(x) - L\right| < \epsilon$ whenever $x > N\;[x < N]$.

But, I think it is also valid when the interval is half-open as in the following: Let $f$ be a function defined on some right-open interval $[a, \infty)$ [left-open interval $(-\infty, a]$]. Then we say the limit of $f(x)$ as $x$ approaches [negative] infinity is $L$, and we write $\lim_{x\to[-]\infty} f(x) = L$ if for every number $\epsilon$, there is a corresponding number $N$ such that $\left|f(x) - L\right| < \epsilon$ whenever $x \geq N\;[x \leq N]$.

So, why the formal definition of limit at infinity does not start with a statement requiring a half-open interval, which is more general?

Is it because people want to match it with the formal definition of limit? I understand that the formal definition of one-sided limit requires an open interval because that is necessary to define a limit at a point $a$. But, such requirement does not exist for limit at infinity, and therefore, why the more general version of half-open interval is not used in the formal definition of limit at infinity?

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There is not really a difference between both approaches: If $f$ is defined on the open interval $(-\infty,a)$, then we may as well consider the restriction to the closed interval $(-\infty,a-1]$ and similarly vice versa. The reason that open interval may be preferred is that the limit requires $f$ to be defined on a topological neighbourhood of $\infty$. A neighbourhood of $\infty$ is a set that contains an open set containing $\infty$ and the basic open sets are open intervals. So the the following definition might be considered "best", but I'm afraid it is way less intuitive for the learner:

Limit At Infinity: Let $f\colon A\to\mathbb R$ be a function where $A$ is a punctured neighbourhood of $\infty$ in the two-point compactification of $\mathbb R$. Then we say ...

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It's great to ask what hypotheses in a theorem or definition are really germane. What is important is that $f$ is defined on some interval $(a,\infty)$ for sufficiently large $a$ so that when we write down the $\epsilon-\delta$ definition, we don't have to worry about pathologies.

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It's not more general. If your function is defined on a closed interval $[a,\infty)$ then it's also defined on an open interval $(b,\infty)$ where $b > a$. And the values of the function on the interval $[a,b]$ don't matter at all when it comes to the limit of the function at $+\infty$.

The question of whether the limit of $f(x)$ as $x\to +\infty$ exists is basically the same as asking whether $f$ can be extended from $(a,\infty)$ to $(a,+\infty]$ (see Wikipedia on the extended real line) such that it's continuous at $+\infty$.

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