Esta é uma lista de limites para as funções matemáticas mais comuns. De referir que a e b são constantes que fazem referência à variável x.
Limites gerais para funções
![{\displaystyle {\text{Se }}\lim _{x\to c}f(x)=L_{1}{\text{ e }}\lim _{x\to c}g(x)=L_{2}{\text{ então:}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f0074d322aa920ad386c60614f6cd24b2fae47e)
![{\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}\,f(x)^{n}=L_{1}^{n}\qquad {\text{ se }}n{\text{ é um inteiro positivo}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e07498222e62c63aab1f995e4174cb4b1e4fb9f7)
![{\displaystyle \lim _{x\to c}\,f(x)^{1 \over n}=L_{1}^{1 \over n}\qquad {\text{ se }}n{\text{ é um inteiro positivo, e se }}n{\text{ é par, então }}L_{1}>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/817d4b7b98d3b4caf123a385c2aa50f44e064c2c)
![{\displaystyle \lim _{x\to c}\,e^{f(x)}=e^{L_{1}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3951ecc227ee98cd68ad239dd7aacbb5397c9a82)
(Regra de l'Hôpital)
Limites de funções gerais
![{\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)
Limites notáveis
![{\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)
[1]
![{\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)
caso
![{\displaystyle \lim _{x\to 0}{\frac {e^{x}-1}{x}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38be7d3073e3ee91b5310a8f509dcfc6189916fb)
Funções 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{ se }}r{\mbox{ é um inteiro positivo}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a1049643a85315b0bafbde9efe7496a28f16b52)
![{\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{se }}r{\text{ é ímpar}}\\+\infty ,&{\text{se }}r{\text{ é par}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f11d5d11860660d8042dd5bb75ba7eaa3c845cac)
Funções logarítmicas e exponenciais
![{\displaystyle {\mbox{Para }}a>1:\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2eef50f5433cb1b398a201eadda499aa5583502)
![{\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{Se }}a<1:\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/622e76b62922d4a57dbbcafe503e6ceda019910b)
![{\displaystyle \lim _{x\to -\infty }a^{x}=\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f573f1a9b7b255c1116ee111dfc915dcbc61a113)
Funções trigonométricas
![{\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)
Se
for expresso em radianos:
![{\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 {\tan x}{x}}=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdb738ab9b4fa01bb32565745a9af6583683c28d)
![{\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)=\pm \infty \qquad {\text{para qualquer inteiro n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0fdc3feefa770a5c099789d27fc410b722b8a900)
![{\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)
Infinitos
![{\displaystyle \lim _{x\to \infty }N/x=0{\text{ para qualquer real }}N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba9c555acf924b57e62e4295fe1e9b0e5e6c9ae)
![{\displaystyle \lim _{x\to \infty }x/N={\begin{cases}\infty ,&N>0\\{\text{não existe}},&N=0\\-\infty ,&N<0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/478e72287c63f296b0a5582cabac057f16f3af84)
![{\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{ para qualquer }}N>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b722763ae392bdc257101baadfb528d7d1f9093)
![{\displaystyle \lim _{x\to \infty }{\sqrt[{x}]{N}}={\begin{cases}1,&N>0\\0,&N=0\\{\text{não existe}},&N<0\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a38df42ce3e9c7e9c4f4f127b66afc415c91589f)
![{\displaystyle \lim _{x\to \infty }{\sqrt[{N}]{x}}=\infty {\text{ para qualquer }}N>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0d3f4353c84a809ea730b5718fd029c6dfc9a92)
![{\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)
Ver também
- ↑ «Confira este exemplo e faça outros com O Monitor». omonitor.io. Consultado em 22 de março de 2016