This is a list of limits for common functions such as elementary functions. In this article, the terms a, b and c are constants with respect to x.
Limits for general functions
Definitions of limits and related concepts
lim x → c f ( x ) = L {\displaystyle \lim _{x\to c}f(x)=L} if and only if ∀ ε > 0 ∃ δ > 0 : 0 < | x − c | < δ ⟹ | f ( x ) − L | < ε {\displaystyle \forall \varepsilon >0\ \exists \delta >0:0<|x-c|<\delta \implies |f(x)-L|<\varepsilon } . This is the (ε, δ)-definition of limit. The limit superior and limit inferior of a sequence are defined as lim sup n → ∞ x n = lim n → ∞ ( sup m ≥ n x m ) {\displaystyle \limsup _{n\to \infty }x_{n}=\lim _{n\to \infty }\left(\sup _{m\geq n}x_{m}\right)} and lim inf n → ∞ x n = lim n → ∞ ( inf m ≥ n x m ) {\displaystyle \liminf _{n\to \infty }x_{n}=\lim _{n\to \infty }\left(\inf _{m\geq n}x_{m}\right)} . A function, f ( x ) {\displaystyle f(x)} , is said to be continuous at a point, c, if lim x → c f ( x ) = f ( c ) . {\displaystyle \lim _{x\to c}f(x)=f(c).}
Operations on a single known limit If lim x → c f ( x ) = L {\displaystyle \lim _{x\to c}f(x)=L} then:
lim x → c [ f ( x ) ± a ] = L ± a {\displaystyle \lim _{x\to c}\,[f(x)\pm a]=L\pm a}
lim x → c a f ( x ) = a L {\displaystyle \lim _{x\to c}\,af(x)=aL}
lim x → c 1 f ( x ) = 1 L {\displaystyle \lim _{x\to c}{\frac {1}{f(x)}}={\frac {1}{L}}} if L is not equal to 0.
lim x → c f ( x ) n = L n {\displaystyle \lim _{x\to c}\,f(x)^{n}=L^{n}} if n is a positive integer
lim x → c f ( x ) 1 n = L 1 n {\displaystyle \lim _{x\to c}\,f(x)^{1 \over n}=L^{1 \over n}} if n is a positive integer, and if n is even, then L > 0. In general, if g(x) is continuous at L and lim x → c f ( x ) = L {\displaystyle \lim _{x\to c}f(x)=L} then
lim x → c g ( f ( x ) ) = g ( L ) {\displaystyle \lim _{x\to c}g\left(f(x)\right)=g(L)}
Operations on two known limits If lim x → c f ( x ) = L 1 {\displaystyle \lim _{x\to c}f(x)=L_{1}} and lim x → c g ( x ) = L 2 {\displaystyle \lim _{x\to c}g(x)=L_{2}} then:
lim x → c [ f ( x ) ± g ( x ) ] = L 1 ± L 2 {\displaystyle \lim _{x\to c}\,[f(x)\pm g(x)]=L_{1}\pm L_{2}}
lim x → c [ f ( x ) g ( x ) ] = L 1 ⋅ L 2 {\displaystyle \lim _{x\to c}\,[f(x)g(x)]=L_{1}\cdot L_{2}}
lim x → c f ( x ) g ( x ) = L 1 L 2 if L 2 ≠ 0 {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {L_{1}}{L_{2}}}\qquad {\text{ if }}L_{2}\neq 0}
Limits involving derivatives or infinitesimal changes In these limits, the infinitesimal change h {\displaystyle h} is often denoted Δ x {\displaystyle \Delta x} or δ x {\displaystyle \delta x} . If f ( x ) {\displaystyle f(x)} is differentiable at x {\displaystyle x} ,
lim h → 0 f ( x + h ) − f ( x ) h = f ′ ( x ) {\displaystyle \lim _{h\to 0}{f(x+h)-f(x) \over h}=f'(x)} . This is the definition of the derivative. All differentiation rules can also be reframed as rules involving limits. For example, if g(x) is differentiable at x,
lim h → 0 f ∘ g ( x + h ) − f ∘ g ( x ) h = f ′ [ g ( x ) ] g ′ ( x ) {\displaystyle \lim _{h\to 0}{f\circ g(x+h)-f\circ g(x) \over h}=f'[g(x)]g'(x)} . This is the chain rule.
lim h → 0 f ( x + h ) g ( x + h ) − f ( x ) g ( x ) h = f ′ ( x ) g ( x ) + f ( x ) g ′ ( x ) {\displaystyle \lim _{h\to 0}{f(x+h)g(x+h)-f(x)g(x) \over h}=f'(x)g(x)+f(x)g'(x)} . This is the product rule.
lim h → 0 ( f ( x + h ) f ( x ) ) 1 / h = exp ( f ′ ( x ) f ( x ) ) {\displaystyle \lim _{h\to 0}\left({\frac {f(x+h)}{f(x)}}\right)^{1/h}=\exp \left({\frac {f'(x)}{f(x)}}\right)}
lim h → 0 ( f ( e h x ) f ( x ) ) 1 / h = exp ( x f ′ ( x ) f ( x ) ) {\displaystyle \lim _{h\to 0}{\left({f(e^{h}x) \over {f(x)}}\right)^{1/h}}=\exp \left({\frac {xf'(x)}{f(x)}}\right)}
If f ( x ) {\displaystyle f(x)} and g ( x ) {\displaystyle g(x)} are differentiable on an open interval containing c, except possibly c itself, and lim x → c f ( x ) = lim x → c g ( x ) = 0 or ± ∞ {\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}g(x)=0{\text{ or }}\pm \infty } , L'Hôpital's rule can be used:
lim x → c f ( x ) g ( x ) = lim x → c f ′ ( x ) g ′ ( x ) {\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}=\lim _{x\to c}{\frac {f'(x)}{g'(x)}}}
Inequalities If f ( x ) ≤ g ( x ) {\displaystyle f(x)\leq g(x)} for all x in an interval that contains c, except possibly c itself, and the limit of f ( x ) {\displaystyle f(x)} and g ( x ) {\displaystyle g(x)} both exist at c, then
lim x → c f ( x ) ≤ lim x → c g ( x ) {\displaystyle \lim _{x\to c}f(x)\leq \lim _{x\to c}g(x)}
If lim x → c f ( x ) = lim x → c h ( x ) = L {\displaystyle \lim _{x\to c}f(x)=\lim _{x\to c}h(x)=L} and f ( x ) ≤ g ( x ) ≤ h ( x ) {\displaystyle f(x)\leq g(x)\leq h(x)} for all x in an open interval that contains c, except possibly c itself,
lim x → c g ( x ) = L . {\displaystyle \lim _{x\to c}g(x)=L.} This is known as the squeeze theorem. This applies even in the cases that f(x) and g(x) take on different values at c, or are discontinuous at c.
Polynomials and functions of the form xa
lim x → c a = a {\displaystyle \lim _{x\to c}a=a}
Polynomials in x
lim x → c x = c {\displaystyle \lim _{x\to c}x=c}
lim x → c ( a x + b ) = a c + b {\displaystyle \lim _{x\to c}(ax+b)=ac+b}
lim x → c x n = c n {\displaystyle \lim _{x\to c}x^{n}=c^{n}} if n is a positive integer
lim x → ∞ x / a = { ∞ , a > 0 does not exist , a = 0 − ∞ , a < 0 {\displaystyle \lim _{x\to \infty }x/a={\begin{cases}\infty ,&a>0\\{\text{does not exist}},&a=0\\-\infty ,&a<0\end{cases}}}
In general, if p ( x ) {\displaystyle p(x)} is a polynomial then, by the continuity of polynomials, lim x → c p ( x ) = p ( c ) {\displaystyle \lim _{x\to c}p(x)=p(c)} This is also true for rational functions, as they are continuous on their domains.
Functions of the form xa
lim x → c x a = c a . {\displaystyle \lim _{x\to c}x^{a}=c^{a}.} In particular,
lim x → ∞ x a = { ∞ , a > 0 1 , a = 0 0 , a < 0 {\displaystyle \lim _{x\to \infty }x^{a}={\begin{cases}\infty ,&a>0\\1,&a=0\\0,&a<0\end{cases}}}
lim x → c x 1 / a = c 1 / a {\displaystyle \lim _{x\to c}x^{1/a}=c^{1/a}} . In particular,
lim x → ∞ x 1 / a = lim x → ∞ x a = ∞ for any a > 0 {\displaystyle \lim _{x\to \infty }x^{1/a}=\lim _{x\to \infty }{\sqrt[{a}]{x}}=\infty {\text{ for any }}a>0}
lim x → 0 + x − n = lim x → 0 + 1 x n = + ∞ {\displaystyle \lim _{x\to 0^{+}}x^{-n}=\lim _{x\to 0^{+}}{\frac {1}{x^{n}}}=+\infty }
lim x → 0 − x − n = lim x → 0 − 1 x n = { − ∞ , if n is odd + ∞ , if n is even {\displaystyle \lim _{x\to 0^{-}}x^{-n}=\lim _{x\to 0^{-}}{\frac {1}{x^{n}}}={\begin{cases}-\infty ,&{\text{if }}n{\text{ is odd}}\\+\infty ,&{\text{if }}n{\text{ is even}}\end{cases}}}
lim x → ∞ a x − 1 = lim x → ∞ a / x = 0 for any real a {\displaystyle \lim _{x\to \infty }ax^{-1}=\lim _{x\to \infty }a/x=0{\text{ for any real }}a}
Exponential functions
Functions of the form ag(x)
lim x → c e x = e c {\displaystyle \lim _{x\to c}e^{x}=e^{c}} , due to the continuity of e x {\displaystyle e^{x}}
lim x → ∞ a x = { ∞ , a > 1 1 , a = 1 0 , 0 < a < 1 {\displaystyle \lim _{x\to \infty }a^{x}={\begin{cases}\infty ,&a>1\\1,&a=1\\0,&0<a<1\end{cases}}}
lim x → ∞ a − x = { 0 , a > 1 1 , a = 1 ∞ , 0 < a < 1 {\displaystyle \lim _{x\to \infty }a^{-x}={\begin{cases}0,&a>1\\1,&a=1\\\infty ,&0<a<1\end{cases}}}
lim x → ∞ a x = lim x → ∞ a 1 / x = { 1 , a > 0 0 , a = 0 does not exist , a < 0 {\displaystyle \lim _{x\to \infty }{\sqrt[{x}]{a}}=\lim _{x\to \infty }{a}^{1/x}={\begin{cases}1,&a>0\\0,&a=0\\{\text{does not exist}},&a<0\end{cases}}}
Functions of the form xg(x)
lim x → ∞ x x = lim x → ∞ x 1 / x = 1 {\displaystyle \lim _{x\to \infty }{\sqrt[{x}]{x}}=\lim _{x\to \infty }{x}^{1/x}=1}
Functions of the form f(x)g(x)
lim x → + ∞ ( x x + k ) x = e − k {\displaystyle \lim _{x\to +\infty }\left({\frac {x}{x+k}}\right)^{x}=e^{-k}}
lim x → 0 ( 1 + x ) 1 x = e {\displaystyle \lim _{x\to 0}\left(1+x\right)^{\frac {1}{x}}=e}
lim x → 0 ( 1 + k x ) m x = e m k {\displaystyle \lim _{x\to 0}\left(1+kx\right)^{\frac {m}{x}}=e^{mk}}
lim x → + ∞ ( 1 + 1 x ) x = e {\displaystyle \lim _{x\to +\infty }\left(1+{\frac {1}{x}}\right)^{x}=e}
lim x → + ∞ ( 1 − 1 x ) x = 1 e {\displaystyle \lim _{x\to +\infty }\left(1-{\frac {1}{x}}\right)^{x}={\frac {1}{e}}}
lim x → + ∞ ( 1 + k x ) m x = e m k {\displaystyle \lim _{x\to +\infty }\left(1+{\frac {k}{x}}\right)^{mx}=e^{mk}}
lim x → 0 ( 1 + a ( e − x − 1 ) ) − 1 x = e a {\displaystyle \lim _{x\to 0}\left(1+a\left({e^{-x}-1}\right)\right)^{-{\frac {1}{x}}}=e^{a}} . This limit can be derived from this limit.
Sums, products and composites
lim x → 0 x e − x = 0 {\displaystyle \lim _{x\to 0}xe^{-x}=0}
lim x → ∞ x e − x = 0 {\displaystyle \lim _{x\to \infty }xe^{-x}=0}
lim x → 0 ( a x − 1 x ) = ln a , {\displaystyle \lim _{x\to 0}\left({\frac {a^{x}-1}{x}}\right)=\ln {a},} for all positive a.
lim x → 0 ( e x − 1 x ) = 1 {\displaystyle \lim _{x\to 0}\left({\frac {e^{x}-1}{x}}\right)=1}
lim x → 0 ( e a x − 1 x ) = a {\displaystyle \lim _{x\to 0}\left({\frac {e^{ax}-1}{x}}\right)=a}
Logarithmic functions
Natural logarithms
lim x → c ln x = ln c {\displaystyle \lim _{x\to c}\ln {x}=\ln c} , due to the continuity of ln x {\displaystyle \ln {x}} . In particular,
lim x → 0 + log x = − ∞ {\displaystyle \lim _{x\to 0^{+}}\log x=-\infty }
lim x → ∞ log x = ∞ {\displaystyle \lim _{x\to \infty }\log x=\infty }
lim x → 1 ln ( x ) x − 1 = 1 {\displaystyle \lim _{x\to 1}{\frac {\ln(x)}{x-1}}=1}
lim x → 0 ln ( x + 1 ) x = 1 {\displaystyle \lim _{x\to 0}{\frac {\ln(x+1)}{x}}=1}
lim x → 0 − ln ( 1 + a ( e − x − 1 ) ) x = a {\displaystyle \lim _{x\to 0}{\frac {-\ln \left(1+a\left({e^{-x}-1}\right)\right)}{x}}=a} . This limit follows from L'Hôpital's rule.
lim x → 0 x ln x = 0 {\displaystyle \lim _{x\to 0}x\ln x=0} , hence lim x → 0 x x = 1 {\displaystyle \lim _{x\to 0}x^{x}=1}
lim x → ∞ ln x x = 0 {\displaystyle \lim _{x\to \infty }{\frac {\ln x}{x}}=0}
Logarithms to arbitrary bases For b > 1,
lim x → 0 + log b x = − ∞ {\displaystyle \lim _{x\to 0^{+}}\log _{b}x=-\infty }
lim x → ∞