[0BK] In the case \(x_ 0ββ\) and \(lββ\), we divide the definition into two conditions: 1
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\(\limsup _{xβ x_ 0} f(x) β€ l\) \(\limsup _{xβ x_ 0} f(x) β₯ l\) |
\(β \varepsilon {\gt}0, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, xβ x_ 0, xβ Iβf(x){\lt}l+\varepsilon \) \(β \varepsilon {\gt}0, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, xβ x_ 0, xβ I, f(x){\gt}l-\varepsilon \) |
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\(\limsup _{xβ x_ 0^+} f(x) β€ l\) \(\limsup _{xβ x_ 0^+} f(x) β₯ l\) |
\(β \varepsilon {\gt}0, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\gt} x_ 0, xβ Iβf(x){\lt}l+\varepsilon \) \(β \varepsilon {\gt}0, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\gt} x_ 0, xβ I, f(x){\gt}l-\varepsilon \) |
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\(\limsup _{xβ x_ 0^-} f(x) β€ l\) \(\limsup _{xβ x_ 0^-} f(x) β₯ l\) |
\(β \varepsilon {\gt}0, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\lt} x_ 0, xβ Iβf(x){\lt}l+\varepsilon \) \(β \varepsilon {\gt}0, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\lt} x_ 0, xβ I, f(x){\gt}l-\varepsilon \) |
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\(\liminf _{xβ x_ 0} f(x) β₯ l\) \(\liminf _{xβ x_ 0} f(x) β€ l\) |
\(β \varepsilon {\gt}0, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, xβ x_ 0, xβ Iβf(x){\gt}l-\varepsilon \) \(β \varepsilon {\gt}0, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, xβ x_ 0, xβ I, f(x){\lt}l+\varepsilon \) |
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\(\liminf _{xβ x_ 0^+} f(x) β₯ l\) \(\liminf _{xβ x_ 0^+} f(x) β€ l\) |
\(β \varepsilon {\gt}0, β πΏ {\gt}0,β x, |x-x_ 0|{\lt}πΏ, x{\gt} x_ 0, xβ Iβf(x){\gt}l-\varepsilon \) \(β \varepsilon {\gt}0, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\gt} x_ 0, xβ I, f(x){\lt}l+\varepsilon \) |
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\(\liminf _{xβ x_ 0^-} f(x) β₯ l\) \(\liminf _{xβ x_ 0^-} f(x) β€ l\) |
\(β \varepsilon {\gt}0, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\lt} x_ 0, xβ Iβf(x){\gt}l-\varepsilon \) \(β \varepsilon {\gt}0, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\lt} x_ 0, xβ I, f(x){\lt}l+\varepsilon \) |
In the case \(x_ 0ββ\) and \(l=Β±β\):
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\(\limsup _{xβ x_ 0} f(x) =β\) |
\(β z, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, xβ x_ 0, xβ I, f(x){\gt}z \) |
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\(\limsup _{xβ x_ 0^+} f(x) =β\) |
\(β z, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\gt} x_ 0, xβ I, f(x){\gt}z \) |
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\(\limsup _{xβ x_ 0^-} f(x) =β\) |
\(β z, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\lt} x_ 0, xβ I, f(x){\gt}z \) |
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\(\limsup _{xβ x_ 0} f(x) =-β\) |
\(β z, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, xβ x_ 0, xβ Iβf(x){\lt}z \) |
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\(\limsup _{xβ x_ 0^+} f(x) =-β\) |
\(β z, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\gt} x_ 0, xβ Iβf(x){\lt}z \) |
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\(\limsup _{xβ x_ 0^-} f(x) =-β\) |
\(β z, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\lt} x_ 0, xβ Iβf(x){\lt}z \) |
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\(\liminf _{xβ x_ 0} f(x) =β\) |
\(β z, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, xβ x_ 0, xβ Iβf(x){\gt}z \) |
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\(\liminf _{xβ x_ 0^+} f(x) =β\) |
\(β z, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\gt} x_ 0, xβ Iβf(x){\gt}z \) |
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\(\liminf _{xβ x_ 0^-} f(x) =β\) |
\(β z, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\lt} x_ 0, xβ Iβf(x){\gt}z \) |
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\(\liminf _{xβ x_ 0} f(x) =-β\) |
\(β z, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, xβ x_ 0, xβ I, f(x){\lt}z \) |
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\(\liminf _{xβ x_ 0^+} f(x) =-β\) |
\(β z, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\gt} x_ 0, xβ I, f(x){\lt}z \) |
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\(\liminf _{xβ x_ 0^-} f(x) =-β\) |
\(β z, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\lt} x_ 0, xβ I, f(x){\lt}z \) |
In the case \(x_ 0=Β±β\) and \(l=Β±β\):
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\(\limsup _{xβ β} f(x) =β\) |
\(β z, β y,β x,x{\gt}y, xβ I, f(x){\gt}z \) |
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\(\limsup _{xβ -β} f(x) =β\) |
\(β z, β y,β x,x{\lt}y, xβ I, f(x){\gt}z \) |
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\(\limsup _{xβ β} f(x) =-β\) |
\(β z, β y,β x,x{\gt}y, xβ Iβf(x){\lt}z \) |
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\(\limsup _{xβ -β} f(x) =-β\) |
\(β z, β y,β x,x{\lt}y, xβ Iβf(x){\lt}z \) |
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\(\liminf _{xβ β} f(x) =β\) |
\(β z, β y,β x,x{\gt}y, xβ Iβf(x){\gt}z \) |
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\(\liminf _{xβ -β} f(x) =β\) |
\(β z, β y,β x,x{\lt}y, xβ Iβf(x){\gt}z \) |
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\(\liminf _{xβ β} f(x) =-β\) |
\(β z, β y,β x,x{\gt}y, xβ I, f(x){\lt}z \) |
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\(\liminf _{xβ -β} f(x) =-β\) |
\(β z, β y,β x,x{\lt}y, xβ I, f(x){\lt}z \) |
In the case \(x_ 0=Β±β\) and \(lββ\):
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\(\limsup _{xβ β} f(x) β€ l\) \(\limsup _{xβ β} f(x) β₯ l\) |
\(β \varepsilon {\gt}0, β y, β x, x{\gt}y, xβ Iβf(x){\lt}l+\varepsilon \) \(β \varepsilon {\gt}0, β y, β x, x{\gt}y, xβ I, f(x){\gt}l-\varepsilon \) |
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\(\limsup _{xβ -β} f(x) β€ l\) \(\limsup _{xβ -β} f(x) β₯ l\) |
\(β \varepsilon {\gt}0, β y, β x, x{\lt}y, xβ Iβf(x){\lt}l+\varepsilon \) \(β \varepsilon {\gt}0, β y, β x, x{\lt}y, xβ I, f(x){\gt}l-\varepsilon \) |
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\(\liminf _{xβ β} f(x) β€ l\) \(\liminf _{xβ β} f(x) β₯ l\) |
\(β \varepsilon {\gt}0, β y, β x, x{\gt}y, xβ I, f(x){\lt}l+\varepsilon \) \(β \varepsilon {\gt}0, β y, β x, x{\gt}y, xβ Iβf(x){\gt}l-\varepsilon \) |
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\(\liminf _{xβ -β} f(x) β€ l\) \(\liminf _{xβ -β} f(x) β₯ l\) |
\(β \varepsilon {\gt}0, β y, β x, x{\lt}y, xβ I, f(x){\lt}l+\varepsilon \) \(β \varepsilon {\gt}0, β y, β x, x{\lt}y, xβ Iβf(x){\gt}l-\varepsilon \) |