[0BH] Sia \(I\) un insieme, \(x_ 0ββ\) punto di accumulazione per \(I\), \(f:Iβ β\) funzione, \(lββ\).
Mettendo insieme tutte le definizioni viste precedentemente, otteniamo queste definizioni di limite.
Nel caso \(x_ 0ββ\) e \(lββ\):
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\(\lim _{xβ x_ 0} f(x) = l\) |
\(β \varepsilon {\gt}0, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, xβ x_ 0, xβ Iβ|f(x)-l|{\lt}\varepsilon \) |
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\(\lim _{xβ x_ 0^+} f(x) = l\) |
\(β \varepsilon {\gt}0, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\gt} x_ 0, xβ Iβ|f(x)-l|{\lt}\varepsilon \) |
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\(\lim _{xβ x_ 0^-} f(x) = l\) |
\(β \varepsilon {\gt}0, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\lt} x_ 0, xβ Iβ|f(x)-l|{\lt}\varepsilon \) |
Sia \(x_ 0ββ\), \(l=Β±β\).
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\(\lim _{xβ x_ 0} f(x) = β\) |
\(β z, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, xβ x_ 0, xβ Iβf(x){\gt}z \) |
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\(\lim _{xβ x_ 0} f(x) = -β\) |
\(β z, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, xβ x_ 0, xβ Iβf(x){\lt}z \) |
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\(\lim _{xβ x_ 0^+} f(x) = β\) |
\(β z, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\gt} x_ 0, xβ Iβf(x){\gt}z \) |
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\(\lim _{xβ x_ 0^+} f(x) = -β\) |
\(β z, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\gt} x_ 0, xβ Iβf(x){\lt}z \) |
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\(\lim _{xβ x_ 0^-} f(x) = β\) |
\(β z, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\lt} x_ 0, xβ Iβf(x){\gt}z \) |
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\(\lim _{xβ x_ 0^-} f(x) = -β\) |
\(β z, β πΏ {\gt}0, β x, |x-x_ 0|{\lt}πΏ, x{\lt} x_ 0, xβ Iβf(x){\lt}z \) |
Sia \(lββ\), \(x_ 0=Β±β\).
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\(\lim _{xβ β} f(x) = l\) |
\(β \varepsilon {\gt}0, β y, β x, x{\gt}y, xβ Iβ|f(x)-l|{\lt}\varepsilon \) |
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\(\lim _{xβ -β} f(x) = l\) |
\(β \varepsilon {\gt}0, β y, β x, x{\lt}y, xβ Iβ|f(x)-l|{\lt}\varepsilon \) |
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\(\lim _{xβ β} f(x) = β\) |
\(β z, β y, β x, x{\gt}y, xβ Iβf(x){\gt}z\) |
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\(\lim _{xβ -β} f(x) = β\) |
\(β z, β y, β x, x{\lt}y, xβ Iβf(x){\gt}z\) |
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\(\lim _{xβ β} f(x) = -β\) |
\(β z, β y, β x, x{\gt}y, xβ Iβf(x){\lt}z\) |
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\(\lim _{xβ -β} f(x) = -β\) |
\(β z, β y, β x, x{\lt}y, xβ Iβf(x){\lt}z\) |