Definisi limit (ε, δ)
Keluarga definisi limit formal

Tidak ada definisi limit yang tunggal - adanya seluruh definisi keluarga. Ini dikarenakan kehadiran takhingga, dan konsep limit "dari sebelah kanan"" dan "dari sebelah kiri". Limit itu sendiri dapat menjadi sebuah nilai terhingga,  \infty , atau  -\infty . Nilai yang mendekati oleh  x juga dapat menjadi nilai terhingga,  \infty , atau  -\infty , dan jika ini merupakan sebuah nilai terhingga, ini dapat mendekati dari kiri atau dari kanan. Biasanya, setiap kombinasinya diberikan definisi itu sendiri, seperti di bawah ini:
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| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| Notasi |  |  |  | Definisi |  |  |  |  |  |  |  |  |  |  | Contoh |
|  \lim _{x\to {\color {Green}c}} |  f(x)= |  {\color {Red}L} |  \iff  |  {\color {Red}\forall \varepsilon >0,} |  {\color {Green}\exists \delta >0,} |  \forall x\in D, |  {\color {Green}c-\delta }< |  x |  <{\color {Green}c+\delta } |  \Rightarrow  |  {\color {Red}L-\varepsilon }< |  f(x) |  <{\color {Red}L+\varepsilon } |  |  \lim _{x\to 0}\sin(x)=0 |
|  \lim _{x\to {\color {Green}c^{+}}} |  f(x)= |  {\color {Red}L} |  \iff  |  {\color {Red}\forall \varepsilon >0,} |  {\color {Green}\exists \delta >0,} |  \forall x\in D, |  {\color {Green}c}< |  x |  <{\color {Green}c+\delta } |  \Rightarrow  |  {\color {Red}L-\varepsilon }< |  f(x) |  <{\color {Red}L+\varepsilon } |  |  \lim _{x\to 0^{+}}x^{2}+\operatorname {sgn}(x)=1 |
|  \lim _{x\to {\color {Green}c^{-}}} |  f(x)= |  {\color {Red}L} |  \iff  |  {\color {Red}\forall \varepsilon >0,} |  {\color {Green}\exists \delta >0,} |  \forall x\in D, |  {\color {Green}c-\delta }< |  x |  <{\color {Green}c} |  \Rightarrow  |  {\color {Red}L-\varepsilon }< |  f(x) |  <{\color {Red}L+\varepsilon } |  |  \lim _{x\to 0^{-}}x^{2}+\operatorname {sgn}(x)=-1 |
|  \lim _{x\to {\color {Green}\infty }} |  f(x)= |  {\color {Red}L} |  \iff  |  {\color {Red}\forall \varepsilon >0,} |  {\color {Green}\exists M>0,} |  \forall x\in D, |  {\color {Green}M}< |  x |  |  \Rightarrow  |  {\color {Red}L-\varepsilon }< |  f(x) |  <{\color {Red}L+\varepsilon } |  |  \lim _{x\to \infty }1/x=0 |
|  \lim _{x\to {\color {Green}-\infty }} |  f(x)= |  {\color {Red}L} |  \iff  |  {\color {Red}\forall \varepsilon >0,} |  {\color {Green}\exists M<0,} |  \forall x\in D, |  |  x |  <{\color {Green}M} |  \Rightarrow  |  {\color {Red}L-\varepsilon }< |  f(x) |  <{\color {Red}L+\varepsilon } |  |  \lim _{x\to -\infty }e^{x}=0 |
|  \lim _{x\to {\color {Green}c}} |  f(x)= |  {\color {Red}\infty } |  \iff  |  {\color {Red}\forall N>0,} |  {\color {Green}\exists \delta >0,} |  \forall x\in D, |  {\color {Green}c-\delta }< |  x |  <{\color {Green}c+\delta } |  \Rightarrow  |  {\color {Red}N}< |  f(x) |  |  |  \lim _{x\to 0}|1/x|=\infty  |
|  \lim _{x\to {\color {Green}c^{+}}} |  f(x)= |  {\color {Red}\infty } |  \iff  |  {\color {Red}\forall N>0,} |  {\color {Green}\exists \delta >0,} |  \forall x\in D, |  {\color {Green}c}< |  x |  <{\color {Green}c+\delta } |  \Rightarrow  |  {\color {Red}N}< |  f(x) |  |  |  \lim _{x\to 0^{+}}1/x=\infty  |
|  \lim _{x\to {\color {Green}c^{-}}} |  f(x)= |  {\color {Red}\infty } |  \iff  |  {\color {Red}\forall N>0,} |  {\color {Green}\exists \delta >0,} |  \forall x\in D, |  {\color {Green}c-\delta }< |  x |  <{\color {Green}c} |  \Rightarrow  |  {\color {Red}N}< |  f(x) |  |  |  \lim _{x\to 0^{-}}-1/x=\infty  |
|  \lim _{x\to {\color {Green}\infty }} |  f(x)= |  {\color {Red}\infty } |  \iff  |  {\color {Red}\forall N>0,} |  {\color {Green}\exists M>0,} |  \forall x\in D, |  {\color {Green}M}< |  x |  |  \Rightarrow  |  {\color {Red}N}< |  f(x) |  |  |  \lim _{x\to \infty }e^{x}=\infty  |
|  \lim _{x\to {\color {Green}-\infty }} |  f(x)= |  {\color {Red}\infty } |  \iff  |  {\color {Red}\forall N>0,} |  {\color {Green}\exists M<0,} |  \forall x\in D, |  |  x |  <{\color {Green}M} |  \Rightarrow  |  {\color {Red}N}< |  f(x) |  |  |  \lim _{x\to -\infty }x^{2}=\infty  |
|  \lim _{x\to {\color {Green}c}} |  f(x)= |  {\color {Red}-\infty } |  \iff  |  {\color {Red}\forall N<0,} |  {\color {Green}\exists \delta >0,} |  \forall x\in D, |  {\color {Green}c-\delta }< |  x |  <{\color {Green}c+\delta } |  \Rightarrow  |  |  f(x) |  <{\color {Red}N} |  |  \lim _{x\to 0}-|1/x|=-\infty  |
|  \lim _{x\to {\color {Green}c^{+}}} |  f(x)= |  {\color {Red}-\infty } |  \iff  |  {\color {Red}\forall N<0,} |  {\color {Green}\exists \delta >0,} |  \forall x\in D, |  {\color {Green}c}< |  x |  <{\color {Green}c+\delta } |  \Rightarrow  |  |  f(x) |  <{\color {Red}N} |  |  \lim _{x\to 0^{+}}\log(x)=-\infty  |
|  \lim _{x\to {\color {Green}c^{-}}} |  f(x)= |  {\color {Red}-\infty } |  \iff  |  {\color {Red}\forall N<0,} |  {\color {Green}\exists \delta >0,} |  \forall x\in D, |  {\color {Green}c-\delta }< |  x |  <{\color {Green}c} |  \Rightarrow  |  |  f(x) |  <{\color {Red}N} |  |  \lim _{x\to 0^{-}}1/x=-\infty  |
|  \lim _{x\to {\color {Green}\infty }} |  f(x)= |  {\color {Red}-\infty } |  \iff  |  {\color {Red}\forall N<0,} |  {\color {Green}\exists M>0,} |  \forall x\in D, |  {\color {Green}M}< |  x |  |  \Rightarrow  |  |  f(x) |  <{\color {Red}N} |  |  \lim _{x\to \infty }-x=-\infty  |
|  \lim _{x\to {\color {Green}-\infty }} |  f(x)= |  {\color {Red}-\infty } |  \iff  |  {\color {Red}\forall N<0,} |  {\color {Green}\exists M<0,} |  \forall x\in D, |  |  x |  <{\color {Green}M} |  \Rightarrow  |  |  f(x) |  <{\color {Red}N} |  |  \lim _{x\to -\infty }x^{3}=-\infty  |