6.5 Upper and lower limits[29P]
From the previous definition we move on to the definitions of “limit superior” \(\limsup \) and “limit inferior” \(\liminf \). The idea is so expressed.
[20F] Let \(I⊂ ℝ\), \(x_ 0∈ \overline{ℝ}\) accumulation point of \(I\), \(f:I→ ℝ\) function. We define
where the first ”inf” (resp. the ”sup”) is performed with respect to the family of all the deleted neighbourhoods \(U\) of \(x_ 0\); and the neighbourhoods will be right or left neighbourhoods if the limit is from right or left.
[20G](Solved on 2022-11-29) Using the properties of \(\inf ,\sup \), we obtain for example these characterizations
and so on. (In this simplified writing, we assume that \(x∈ I\)).
[20N]You have \(𝛼=\limsup _{x→ x_ 0} f(x)\) if and only if
[0BK] In the case \(x_ 0∈ℝ\) and \(l∈ℝ\), we divide the definition into two conditions: 1
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\(\limsup _{x→ x_ 0} f(x) ≤ l\) \(\limsup _{x→ x_ 0} f(x) ≥ l\) |
\(∀ \varepsilon {\gt}0, ∃ 𝛿 {\gt}0, ∀ x, |x-x_ 0|{\lt}𝛿, x≠ x_ 0, x∈ I⇒f(x){\lt}l+\varepsilon \) \(∀ \varepsilon {\gt}0, ∀ 𝛿 {\gt}0, ∃ x, |x-x_ 0|{\lt}𝛿, x≠ x_ 0, x∈ I, f(x){\gt}l-\varepsilon \) |
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\(\limsup _{x→ x_ 0^+} f(x) ≤ l\) \(\limsup _{x→ x_ 0^+} f(x) ≥ l\) |
\(∀ \varepsilon {\gt}0, ∃ 𝛿 {\gt}0, ∀ x, |x-x_ 0|{\lt}𝛿, x{\gt} x_ 0, x∈ I⇒f(x){\lt}l+\varepsilon \) \(∀ \varepsilon {\gt}0, ∀ 𝛿 {\gt}0, ∃ x, |x-x_ 0|{\lt}𝛿, x{\gt} x_ 0, x∈ I, f(x){\gt}l-\varepsilon \) |
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\(\limsup _{x→ x_ 0^-} f(x) ≤ l\) \(\limsup _{x→ x_ 0^-} f(x) ≥ l\) |
\(∀ \varepsilon {\gt}0, ∃ 𝛿 {\gt}0, ∀ x, |x-x_ 0|{\lt}𝛿, x{\lt} x_ 0, x∈ I⇒f(x){\lt}l+\varepsilon \) \(∀ \varepsilon {\gt}0, ∀ 𝛿 {\gt}0, ∃ x, |x-x_ 0|{\lt}𝛿, x{\lt} x_ 0, x∈ I, f(x){\gt}l-\varepsilon \) |
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\(\liminf _{x→ x_ 0} f(x) ≥ l\) \(\liminf _{x→ x_ 0} f(x) ≤ l\) |
\(∀ \varepsilon {\gt}0, ∃ 𝛿 {\gt}0, ∀ x, |x-x_ 0|{\lt}𝛿, x≠ x_ 0, x∈ I⇒f(x){\gt}l-\varepsilon \) \(∀ \varepsilon {\gt}0, ∀ 𝛿 {\gt}0, ∃ x, |x-x_ 0|{\lt}𝛿, x≠ x_ 0, x∈ I, f(x){\lt}l+\varepsilon \) |
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\(\liminf _{x→ x_ 0^+} f(x) ≥ l\) \(\liminf _{x→ x_ 0^+} f(x) ≤ l\) |
\(∀ \varepsilon {\gt}0, ∃ 𝛿 {\gt}0,∀ x, |x-x_ 0|{\lt}𝛿, x{\gt} x_ 0, x∈ I⇒f(x){\gt}l-\varepsilon \) \(∀ \varepsilon {\gt}0, ∀ 𝛿 {\gt}0, ∃ x, |x-x_ 0|{\lt}𝛿, x{\gt} x_ 0, x∈ I, f(x){\lt}l+\varepsilon \) |
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\(\liminf _{x→ x_ 0^-} f(x) ≥ l\) \(\liminf _{x→ x_ 0^-} f(x) ≤ l\) |
\(∀ \varepsilon {\gt}0, ∃ 𝛿 {\gt}0, ∀ x, |x-x_ 0|{\lt}𝛿, x{\lt} x_ 0, x∈ I⇒f(x){\gt}l-\varepsilon \) \(∀ \varepsilon {\gt}0, ∀ 𝛿 {\gt}0, ∃ x, |x-x_ 0|{\lt}𝛿, x{\lt} x_ 0, x∈ I, f(x){\lt}l+\varepsilon \) |
In the case \(x_ 0∈ℝ\) and \(l=±∞\):
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\(\limsup _{x→ x_ 0} f(x) =∞\) |
\(∀ z, ∀ 𝛿 {\gt}0, ∃ x, |x-x_ 0|{\lt}𝛿, x≠ x_ 0, x∈ I, f(x){\gt}z \) |
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\(\limsup _{x→ x_ 0^+} f(x) =∞\) |
\(∀ z, ∀ 𝛿 {\gt}0, ∃ x, |x-x_ 0|{\lt}𝛿, x{\gt} x_ 0, x∈ I, f(x){\gt}z \) |
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\(\limsup _{x→ x_ 0^-} f(x) =∞\) |
\(∀ z, ∀ 𝛿 {\gt}0, ∃ x, |x-x_ 0|{\lt}𝛿, x{\lt} x_ 0, x∈ I, f(x){\gt}z \) |
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\(\limsup _{x→ x_ 0} f(x) =-∞\) |
\(∀ z, ∃ 𝛿 {\gt}0, ∀ x, |x-x_ 0|{\lt}𝛿, x≠ x_ 0, x∈ I⇒f(x){\lt}z \) |
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\(\limsup _{x→ x_ 0^+} f(x) =-∞\) |
\(∀ z, ∃ 𝛿 {\gt}0, ∀ x, |x-x_ 0|{\lt}𝛿, x{\gt} x_ 0, x∈ I⇒f(x){\lt}z \) |
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\(\limsup _{x→ x_ 0^-} f(x) =-∞\) |
\(∀ z, ∃ 𝛿 {\gt}0, ∀ x, |x-x_ 0|{\lt}𝛿, x{\lt} x_ 0, x∈ I⇒f(x){\lt}z \) |
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\(\liminf _{x→ x_ 0} f(x) =∞\) |
\(∀ z, ∃ 𝛿 {\gt}0, ∀ x, |x-x_ 0|{\lt}𝛿, x≠ x_ 0, x∈ I⇒f(x){\gt}z \) |
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\(\liminf _{x→ x_ 0^+} f(x) =∞\) |
\(∀ z, ∃ 𝛿 {\gt}0, ∀ x, |x-x_ 0|{\lt}𝛿, x{\gt} x_ 0, x∈ I⇒f(x){\gt}z \) |
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\(\liminf _{x→ x_ 0^-} f(x) =∞\) |
\(∀ z, ∃ 𝛿 {\gt}0, ∀ x, |x-x_ 0|{\lt}𝛿, x{\lt} x_ 0, x∈ I⇒f(x){\gt}z \) |
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\(\liminf _{x→ x_ 0} f(x) =-∞\) |
\(∀ z, ∀ 𝛿 {\gt}0, ∃ x, |x-x_ 0|{\lt}𝛿, x≠ x_ 0, x∈ I, f(x){\lt}z \) |
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\(\liminf _{x→ x_ 0^+} f(x) =-∞\) |
\(∀ z, ∀ 𝛿 {\gt}0, ∃ x, |x-x_ 0|{\lt}𝛿, x{\gt} x_ 0, x∈ I, f(x){\lt}z \) |
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\(\liminf _{x→ x_ 0^-} f(x) =-∞\) |
\(∀ z, ∀ 𝛿 {\gt}0, ∃ x, |x-x_ 0|{\lt}𝛿, x{\lt} x_ 0, x∈ I, f(x){\lt}z \) |
In the case \(x_ 0=±∞\) and \(l=±∞\):
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\(\limsup _{x→ ∞} f(x) =∞\) |
\(∀ z, ∀ y,∃ x,x{\gt}y, x∈ I, f(x){\gt}z \) |
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\(\limsup _{x→ -∞} f(x) =∞\) |
\(∀ z, ∀ y,∃ x,x{\lt}y, x∈ I, f(x){\gt}z \) |
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\(\limsup _{x→ ∞} f(x) =-∞\) |
\(∀ z, ∃ y,∀ x,x{\gt}y, x∈ I⇒f(x){\lt}z \) |
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\(\limsup _{x→ -∞} f(x) =-∞\) |
\(∀ z, ∃ y,∀ x,x{\lt}y, x∈ I⇒f(x){\lt}z \) |
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\(\liminf _{x→ ∞} f(x) =∞\) |
\(∀ z, ∃ y,∀ x,x{\gt}y, x∈ I⇒f(x){\gt}z \) |
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\(\liminf _{x→ -∞} f(x) =∞\) |
\(∀ z, ∃ y,∀ x,x{\lt}y, x∈ I⇒f(x){\gt}z \) |
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\(\liminf _{x→ ∞} f(x) =-∞\) |
\(∀ z, ∀ y,∃ x,x{\gt}y, x∈ I, f(x){\lt}z \) |
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\(\liminf _{x→ -∞} f(x) =-∞\) |
\(∀ z, ∀ y,∃ x,x{\lt}y, x∈ I, f(x){\lt}z \) |
In the case \(x_ 0=±∞\) and \(l∈ℝ\):
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\(\limsup _{x→ ∞} f(x) ≤ l\) \(\limsup _{x→ ∞} f(x) ≥ l\) |
\(∀ \varepsilon {\gt}0, ∃ y, ∀ x, x{\gt}y, x∈ I⇒f(x){\lt}l+\varepsilon \) \(∀ \varepsilon {\gt}0, ∀ y, ∃ x, x{\gt}y, x∈ I, f(x){\gt}l-\varepsilon \) |
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\(\limsup _{x→ -∞} f(x) ≤ l\) \(\limsup _{x→ -∞} f(x) ≥ l\) |
\(∀ \varepsilon {\gt}0, ∃ y, ∀ x, x{\lt}y, x∈ I⇒f(x){\lt}l+\varepsilon \) \(∀ \varepsilon {\gt}0, ∀ y, ∃ x, x{\lt}y, x∈ I, f(x){\gt}l-\varepsilon \) |
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\(\liminf _{x→ ∞} f(x) ≤ l\) \(\liminf _{x→ ∞} f(x) ≥ l\) |
\(∀ \varepsilon {\gt}0, ∀ y, ∃ x, x{\gt}y, x∈ I, f(x){\lt}l+\varepsilon \) \(∀ \varepsilon {\gt}0, ∃ y, ∀ x, x{\gt}y, x∈ I⇒f(x){\gt}l-\varepsilon \) |
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\(\liminf _{x→ -∞} f(x) ≤ l\) \(\liminf _{x→ -∞} f(x) ≥ l\) |
\(∀ \varepsilon {\gt}0, ∀ y, ∃ x, x{\lt}y, x∈ I, f(x){\lt}l+\varepsilon \) \(∀ \varepsilon {\gt}0, ∃ y, ∀ x, x{\lt}y, x∈ I⇒f(x){\gt}l-\varepsilon \) |
[0BM]Note that
and
[0BN]Note that if you replace \(f↦ -f\), \(l↦ -l\), you pass from the definitions of \(\limsup \) to those of \(\liminf \) (and vice versa). Another symmetry is achieved by switching \(x_ 0 → -x_ 0\) and right and left neighbourhoods/limits.
- E199
[0BP] Let \(A_ 1,A_ 2\ldots \) be sets , for \(n∈ℕ\); let \(X=⋃_ n A_ n\). We define the characteristic function \({\mathbb 1}_ A: X→ℝ\) as
\[ {\mathbb 1}_ A(x)= \begin{cases} 1 & \text{if}~ x∈ A\\ {} 0 & \text{if}~ x∉ A \end{cases} ~ . \]We will use the definitions \(\limsup _{n} A_ n\) and \(\liminf _{n} A_ n\) seen in eqn. ?? and ??. You have
\begin{eqnarray} \label{eq:limsup_ insiemi_ 1} {\mathbb 1}_{(\limsup _{n} A_ n)} & =& \limsup _{n} {\mathbb 1}_{A_ n} ~ ~ ,\\ {} \label{eq:liminf_ insiemi_ 1} {\mathbb 1}_{(\liminf _{n} A_ n)} & =& \liminf _{n} {\mathbb 1}_{A_ n} ~ ~ . \end{eqnarray}- E199
[0BQ] We fix a real valued sequence \(a_ n\). Now consider the definition of 192 setting \(I=ℕ\) and \(x_ 0=∞\), so that neighborhoods of \(x_ 0\) are sets containing \([n,∞)=\{ m∈ℕ:m≥ n\} \); with these assumptions show that you have
\begin{align} \limsup _{n→ ∞} a_ n =& \inf _ n \sup _{m≥ n} a_ n= \lim _{n→∞} \sup _{m≥ n} a_ n~ ~ , \nonumber \\ \liminf _{n→ ∞} a_ n =& \sup _ n \inf _{m≥ n} a_ n= \lim _{n→∞} \inf _{m≥ n} a_ n~ . \label{eq:def_ limsup_ liminf_ succ}, \end{align}- E199
[29R]Prerequisites:192,175,53,237,3.Difficulty:*.(Proposed on 2022-11-24)
Let \(I⊂ ℝ\), \(x_ 0∈ \overline{ℝ}\) accumulation point of \(I\), \(f:I→ ℝ\) function. As in 175 \({\mathcal F}\) all the neighbourhoods of \(x_ 0\) with associated the filtering ordering
\[ U,V∈ {{\mathcal F}}~ ~ , U≤ V \iff U⊇ V\quad . \]Let
\[ s,i : {\mathcal F}→ℝ~ ~ ,~ ~ s(U) = \sup _{x∈ U∩ I} f(x)~ ~ ,~ ~ i(U) = \inf _{x∈ U∩ I} f(x) \]note that these are monotonic functions, and show that 2
\begin{align} \limsup _{x→ x_ 0} f(x) {\stackrel{.}{=}}\inf _{U∈ {\mathcal F}} s(U) = \lim _{U∈ {\mathcal F}} s(U) \label{eq:limsup_ R_ F} \\ \liminf _{x→ x_ 0} f(x) {\stackrel{.}{=}}\sup _{U ∈ {\mathcal F}} i(U) = \lim _{U ∈ {\mathcal F}} i(U) \label{eq:liminf_ R_ F} \end{align}where the limits are defined in 237.
- E199
[29S]Prerequisites:3.(Solved on 2022-11-24)
Let \(I⊂ ℝ\), \(x_ 0∈ \overline{ℝ}\) accumulation point of \(I\), and \(f,g:I→ ℝ\) functions. Prove that
\[ \limsup _{x→ x_ 0} \big( f(x) + g(x) \big) \le \limsup _{x→ x_ 0} f(x) + \limsup _{x→ x_ 0} g(x)\quad . \]- E199
[29T]Let \(I⊂ ℝ\), \(x_ 0∈ \overline{ℝ}\) accumulation point of \(I\), \(f:I→ ℝ\) function. Let \(r{\gt}0,t∈ℝ,𝜌{\lt}0\); show that
\begin{align*} \limsup _{x→ x_ 0} (f(x) +t) = t+\limsup _{x→ x_ 0}f(x)~ ~ ,~ ~ \limsup _{x→ x_ 0}(r f(x)) = r \limsup _{x→ x_ 0} f(x)~ ~ ,~ ~ \\ \limsup _{x→ x_ 0}(𝜌 f(x)) = 𝜌 \liminf _{x→ x_ 0} f(x)~ ~ . \end{align*}
Other exercises on limits of sequences can be found in Sec. 7.1.