lim x → a [ f ( x ) + g ( x ) ] = lim x → a f ( x ) + lim x → a g ( x ) {\displaystyle \lim _{x\rightarrow a}\left[f(x)+g(x)\right]=\lim _{x\rightarrow a}f(x)+\lim _{x\rightarrow a}g(x)} lim x → a [ f ( x ) − g ( x ) ] = lim x → a f ( x ) − lim x → a g ( x ) {\displaystyle \lim _{x\rightarrow a}\left[f(x)-g(x)\right]=\lim _{x\rightarrow a}f(x)-\lim _{x\rightarrow a}g(x)} lim x → a [ c f ( x ) ] = c lim x → a f ( x ) {\displaystyle \lim _{x\rightarrow a}\left[cf(x)\right]=c\lim _{x\rightarrow a}f(x)} lim x → a [ f ( x ) g ( x ) ] = lim x → a f ( x ) ⋅ lim x → a g ( x ) {\displaystyle \lim _{x\rightarrow a}\left[f(x)g(x)\right]=\lim _{x\rightarrow a}f(x)\cdot \lim _{x\rightarrow a}g(x)} lim x → a [ f ( x ) g ( x ) ] = lim x → a f ( x ) lim x → a g ( x ) if lim x → a g ( x ) ≠ 0 {\displaystyle \lim _{x\rightarrow a}\left[{\frac {f(x)}{g(x)}}\right]={\frac {\lim _{x\rightarrow a}f(x)}{\lim _{x\rightarrow a}g(x)}}{\mbox{ if }}\lim _{x\rightarrow a}g(x)\neq 0} lim x → a [ f ( x ) ] n = [ lim x → a f ( x ) ] n {\displaystyle \lim _{x\rightarrow a}\left[f(x)\right]^{n}=\left[\lim _{x\rightarrow a}f(x)\right]^{n}} lim x → a f ( x ) n = lim x → a f ( x ) n {\displaystyle \lim _{x\rightarrow a}{\sqrt[{n}]{f(x)}}={\sqrt[{n}]{\lim _{x\rightarrow a}f(x)}}}
lim x → a c = c {\displaystyle \lim _{x\rightarrow a}c=c} lim x → a x = a {\displaystyle \lim _{x\rightarrow a}x=a} lim x → a x n = a n {\displaystyle \lim _{x\rightarrow a}x^{n}=a^{n}} lim x → a x n = a n {\displaystyle \lim _{x\rightarrow a}{\sqrt[{n}]{x}}={\sqrt[{n}]{a}}}
If f {\displaystyle f\!} is a polynomial or a rational function and a {\displaystyle a\!} is in the domain of f {\displaystyle f\!} , then lim x → a f ( x ) = f ( a ) {\displaystyle \lim _{x\rightarrow a}f(x)=f(a)}
The derivative of a function f {\displaystyle f\!} at a number a {\displaystyle a\!} , denoted by f ′ ( a ) {\displaystyle f'(a)\!} is f ′ ( a ) = lim h → 0 f ( a + h ) − f ( a ) h {\displaystyle f'(a)=\lim _{h\rightarrow 0}{\frac {f(a+h)-f(a)}{h}}} if this limit exists.
![{\displaystyle \lim _{x\rightarrow a}\left[f(x)+g(x)\right]=\lim _{x\rightarrow a}f(x)+\lim _{x\rightarrow a}g(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/36a04a5a8bce61357133c78643293c4390893d2f)
![{\displaystyle \lim _{x\rightarrow a}\left[f(x)-g(x)\right]=\lim _{x\rightarrow a}f(x)-\lim _{x\rightarrow a}g(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b8333906b33d5032d180115cc39b7027688497c)
![{\displaystyle \lim _{x\rightarrow a}\left[cf(x)\right]=c\lim _{x\rightarrow a}f(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eab06e1a41106f180c58e2d32e90ad08c3d30557)
![{\displaystyle \lim _{x\rightarrow a}\left[f(x)g(x)\right]=\lim _{x\rightarrow a}f(x)\cdot \lim _{x\rightarrow a}g(x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e6df8850c191a828f03270853f2951583ecefdc8)
![{\displaystyle \lim _{x\rightarrow a}\left[{\frac {f(x)}{g(x)}}\right]={\frac {\lim _{x\rightarrow a}f(x)}{\lim _{x\rightarrow a}g(x)}}{\mbox{ if }}\lim _{x\rightarrow a}g(x)\neq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5756d0a9a46ec9a6fb0051a8d25733e51da30778)
![{\displaystyle \lim _{x\rightarrow a}\left[f(x)\right]^{n}=\left[\lim _{x\rightarrow a}f(x)\right]^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f1522761f72335974e28687a398de51a8cda831)
![{\displaystyle \lim _{x\rightarrow a}{\sqrt[{n}]{f(x)}}={\sqrt[{n}]{\lim _{x\rightarrow a}f(x)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71c16d59b49bcad74d5235b84fb1e2a593f9ad3e)
![{\displaystyle \lim _{x\rightarrow a}{\sqrt[{n}]{x}}={\sqrt[{n}]{a}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe1f029766c05ed9ae8c81f39e7a31e149ff77cb)
is a polynomial or a rational function and 
is in the domain of , then
at a number , denoted by 
is
if this limit exists.
