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A continuació es mostra una llista de límits de funcions comunes. Noti's que a i b són constant respecte x.
Propietats generals de límits
![{\displaystyle \lim _{x\to c}\,[f(x)\pm g(x)]=L_{1}\pm L_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cb00a9174f995bae626da947886d0229d14b275)
![{\displaystyle \lim _{x\to c}\,[f(x)g(x)]=L_{1}\times L_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d8693d558a7dfc0f5f0c5900a05dd950f6f7f1)
![{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {L_{1}}{L_{2}}}\qquad {\text{ si }}L_{2}\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e8b1ea8f30e2bc53d23060936f3fe42f3e20566c)
![{\displaystyle \lim _{x\to c}\,f(x)^{n}=L_{1}^{n}\qquad {\text{ si }}n{\text{ és un enter positiu}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46fe91d48be741b67d81bb061f29f9e541d69588)
![{\displaystyle \lim _{x\to c}\,f(x)^{1 \over n}=L_{1}^{1 \over n}\qquad {\text{ si }}n{\text{ és un enter positiu, i si }}n{\text{ és parell, llavors }}L_{1}>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0948338142ceb4c948262e7604e73b36ed2f7966)
(Regla de l'Hôpital)
Límits de funcions generals
![{\displaystyle \lim _{h\to 0}{f(x+h)-f(x) \over h}=f'(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6148d980c98953dbd86c98078c4d653bedc41af)
![{\displaystyle \lim _{h\to 0}\left({\frac {f(x+h)}{f(x)}}\right)^{\frac {1}{h}}=\exp \left({\frac {f'(x)}{f(x)}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a11d7e228e328f7c206e0bdf7a26f2b8684c7d4c)
![{\displaystyle \lim _{h\to 0}{\left({f(x(1+h)) \over {f(x)}}\right)^{1 \over {h}}}=\exp \left({\frac {xf'(x)}{f(x)}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/942c956bb9d9415e620bf752f4e37d9b8cbaaf4e)
Límits notables especials
![{\displaystyle \lim _{x\to +\infty }\left(1+{\frac {k}{x}}\right)^{mx}=e^{mk}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/add4edf8e48c9e29210a0df91c0c28a4c6fb3a80)
![{\displaystyle \lim _{x\to +\infty }\left(1+{\frac {1}{x}}\right)^{x}=e}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98f1dae2cd43f2be63ac43cf61e39ab5ed028179)
![{\displaystyle \lim _{x\to +\infty }\left(1-{\frac {1}{x}}\right)^{x}={\frac {1}{e}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9db2ba69ebe50f31110495ba9c7d0c45e2754281)
![{\displaystyle \lim _{x\to +\infty }\left(1+{\frac {k}{x}}\right)^{x}=e^{k}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d33901e97ce95f7ea5f47ffdd47d0c30b1ec47d3)
![{\displaystyle \lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}=e}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e67d9f7e2588c9b3d418f1107e9ea27b8f330ed)
![{\displaystyle \lim _{n\to \infty }\,2^{n}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+{\text{...}}+{\sqrt {2}}}}}}}} _{n}=\pi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/15782054b42f9fbb4be35ecb85a1de97abfc6709)
![{\displaystyle \lim _{x\to 0}\left({\frac {a^{x}-1}{x}}\right)=\ln {a},\qquad \forall ~a>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7c664557223e86935a2690b4785974124d1c7b6c)
Funcions simples
![{\displaystyle \lim _{x\to c}a=a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f4aaff459f69192f8dc759314582c6751a41363)
![{\displaystyle \lim _{x\to c}x=c}](https://wikimedia.org/api/rest_v1/media/math/render/svg/337e2c532e2cedd5b0d67bd903d39cd1cbacaf77)
![{\displaystyle \lim _{x\to c}ax+b=ac+b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d30f7a105c48fca95512fc7752493eb7dfa6d29c)
![{\displaystyle \lim _{x\to c}x^{r}=c^{r}\qquad {\mbox{ si }}r{\mbox{ és un enter positiu}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a00e59bca17e29f408508e04d038a047fac4c788)
![{\displaystyle \lim _{x\to 0^{+}}{\frac {1}{x^{r}}}=+\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/97a337010a50e6173a4b1c1b5953ac7ed9ed4bab)
![{\displaystyle \lim _{x\to 0^{-}}{\frac {1}{x^{r}}}={\begin{cases}-\infty ,&{\text{si }}r{\text{ és senar}}\\+\infty ,&{\text{si }}r{\text{ és parell}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1782eeb9d793c7582472a772244dedd2bb203aaf)
Funcions logarítmiques i exponencials
![{\displaystyle \lim _{x\to 1}{\frac {\ln(x)}{x-1}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31e8144e6486167e696a70b49e1ef2ff4019cbec)
o:
![{\displaystyle \lim _{x\to 0}{\frac {\ln(x+1)}{x}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bc1b1d0b70b18bd95c186e02fc7fdaa96645002)
![{\displaystyle {\mbox{Per }}a>1:\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3eda7cc3b5f31fb396c8e45ae238af5f140483ce)
![{\displaystyle \lim _{x\to 0^{+}}\log _{a}x=-\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/72808181c55d9096253749e5968f71c47c87bd9b)
![{\displaystyle \lim _{x\to \infty }\log _{a}x=\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bac08a36cc877e299461440d1c9c08d88373f041)
![{\displaystyle \lim _{x\to -\infty }a^{x}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e5300504623b781a3067f7f6a88dfb5bf0ffb46f)
![{\displaystyle {\mbox{Si }}a<1:\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2dbfb7338de52c0b5acd6aa96a24cf0bd950b348)
![{\displaystyle \lim _{x\to -\infty }a^{x}=\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f573f1a9b7b255c1116ee111dfc915dcbc61a113)
Funcions trigonomètriques
![{\displaystyle \lim _{x\to a}\sin x=\sin a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/04bf4a31eeee313433f14413d9e3c441d69576a9)
![{\displaystyle \lim _{x\to a}\cos x=\cos a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/233fccfb3bbfdad201662ecc5dce951fd4baf7b9)
Si
està expressat en radiants:
![{\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0f2fb52f5211c7b7aa69d9e75195afaab5b9d5b1)
![{\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e68c22c5c70aa65f792c06b2cf2fb6e1ecfe16b)
![{\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x^{2}}}={\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/580636aba444eda89b876cfc9e36e1f31d40e771)
![{\displaystyle \lim _{x\to n^{\pm }}\tan \left(\pi x+{\frac {\pi }{2}}\right)=\mp \infty \qquad {\text{per tot enter }}n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d6ce4a7d887ace83c5582622d939261ff0e500f)
![{\displaystyle \lim _{x\to 0}{\frac {\sin ax}{x}}=a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/415364a0432ca572cee3f3ec1fa393bc925f9124)
![{\displaystyle \lim _{x\to 0}{\frac {\sin ax}{\sin bx}}={\frac {a}{b}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a464ba49453137dbac7148ded939c3a2ed945e1e)
Límits a infinit
![{\displaystyle \lim _{x\to \infty }N/x=0{\text{ per tot nombre real }}N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ceb0d08460530318165e8cf8dc3683134c76eb59)
![{\displaystyle \lim _{x\to \infty }x/N={\begin{cases}\infty ,&N>0\\{\text{no existeix}},&N=0\\-\infty ,&N<0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db98b3a84093bad47aadfb3b00256ccbf046a3e0)
![{\displaystyle \lim _{x\to \infty }x^{N}={\begin{cases}\infty ,&N>0\\1,&N=0\\0,&N<0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d65f9d24a6d8cf90b0d955281e7ad9c6f78a182a)
![{\displaystyle \lim _{x\to \infty }N^{x}={\begin{cases}\infty ,&N>1\\1,&N=1\\0,&0<N<1\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4095d481ad66b14be21345ba0bcfb1cc9c102e)
![{\displaystyle \lim _{x\to \infty }N^{-x}=\lim _{x\to \infty }1/N^{x}=0{\text{ per tot }}N>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16883980930bf0a64a278420e7072c0acb9093e7)
![{\displaystyle \lim _{x\to \infty }{\sqrt[{x}]{N}}={\begin{cases}1,&N>0\\0,&N=0\\{\text{no existeix}},&N<0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/75b17ad323ea1438561e93667ee71c0f7b2d002c)
![{\displaystyle \lim _{x\to \infty }{\sqrt[{N}]{x}}=\infty {\text{ per tot }}N>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/366b6eec006b83857ddf91fd9c636d1f94064e15)
![{\displaystyle \lim _{x\to \infty }\log x=\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a3dc39adf241c3465eb63b5e002296b7d0c57e)
![{\displaystyle \lim _{x\to 0^{+}}\log x=-\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb7a454bbcf3fafcf6ea82fdcd1b8346c5c0d1a7)
Referències
Bibliografia
- Ortega Aramburu, Joaquín M. Introducció a l'anàlisi matemàtica. Bellaterra: Servei de Publicacions de la Universitat Autònoma de Barcelona, 2002. ISBN 8449022711.
- Perelló, Carles. Càlcul infinitesimal. Barcelona: Enciclopèdia Catalana, 1994. ISBN 84-7739-518-7.
- Servi, L.D. «Nested square roots of 2». Amer. Math. Monthly, 110, núm. 4, 2003, pàg. 326-330.