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 [ edit ] 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 [ edit ] 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} [ 1] [ 2] [ 3] lim x → c 1 f ( x ) = 1 L {\displaystyle \lim _{x\to c}{\frac {1}{f(x)}}={\frac {1}{L}}} [ 4] 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[ 1] [ 2] [ 3] 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.[ 1] [ 3] 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)} [ 1] [ 2] Operations on two known limits [ edit ] 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}} [ 1] [ 2] [ 3] 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}} [ 1] [ 2] [ 3] 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} [ 1] [ 2] [ 3] Limits involving derivatives or infinitesimal changes [ edit ] 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)}}} [ 2] 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[ 5] 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 .[ 1] [ 2] This applies even in the cases that f (x ) and g (x ) take on different values at c , or are discontinuous at c .
lim x → c a = a {\displaystyle \lim _{x\to c}a=a} [ 1] [ 2] [ 3] lim x → c x = c {\displaystyle \lim _{x\to c}x=c} [ 1] [ 2] [ 3] 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[ 5] 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,[ 5] 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 .[ 5]
lim x → c x a = c a . {\displaystyle \lim _{x\to c}x^{a}=c^{a}.} [ 5] 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}} .[ 5] 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} [ 6] 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 [ edit ] 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}}} [ 6] 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}}} 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} lim x → + ∞ ( x x + k ) x = e − k {\displaystyle \lim _{x\to +\infty }\left({\frac {x}{x+k}}\right)^{x}=e^{-k}} [ 2] lim x → 0 ( 1 + x ) 1 x = e {\displaystyle \lim _{x\to 0}\left(1+x\right)^{\frac {1}{x}}=e} [ 2] 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} [ 7] 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}} [ 6] 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[ edit ] 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 .[ 4] [ 7] 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 [ edit ] 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} [ 7] 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} [ 6] Logarithms to arbitrary bases [ edit ] For b > 1,
lim x → 0 + log b x = − ∞ {\displaystyle \lim _{x\to 0^{+}}\log _{b}x=-\infty } lim x → ∞ log b x = ∞ {\displaystyle \lim _{x\to \infty }\log _{b}x=\infty } For b < 1,
lim x → 0 + log b x = ∞ {\displaystyle \lim _{x\to 0^{+}}\log _{b}x=\infty } lim x → ∞ log b x = − ∞ {\displaystyle \lim _{x\to \infty }\log _{b}x=-\infty } Both cases can be generalized to:
lim x → 0 + log b x = − F ( b ) ∞ {\displaystyle \lim _{x\to 0^{+}}\log _{b}x=-F(b)\infty } lim x → ∞ log b x = F ( b ) ∞ {\displaystyle \lim _{x\to \infty }\log _{b}x=F(b)\infty } where F ( x ) = 2 H ( x − 1 ) − 1 {\displaystyle F(x)=2H(x-1)-1} and H ( x ) {\displaystyle H(x)} is the Heaviside step function
Trigonometric functions [ edit ] If x {\displaystyle x} is expressed in radians:
lim x → a sin x = sin a {\displaystyle \lim _{x\to a}\sin x=\sin a} lim x → a cos x = cos a {\displaystyle \lim _{x\to a}\cos x=\cos a} These limits both follow from the continuity of sin and cos.
lim x → 0 sin x x = 1 {\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=1} .[ 7] [ 8] Or, in general, lim x → 0 sin a x a x = 1 {\displaystyle \lim _{x\to 0}{\frac {\sin ax}{ax}}=1} , for a not equal to 0. lim x → 0 sin a x x = a {\displaystyle \lim _{x\to 0}{\frac {\sin ax}{x}}=a} lim x → 0 sin a x b x = a b {\displaystyle \lim _{x\to 0}{\frac {\sin ax}{bx}}={\frac {a}{b}}} , for b not equal to 0. lim x → ∞ x sin ( 1 x ) = 1 {\displaystyle \lim _{x\to \infty }x\sin \left({\frac {1}{x}}\right)=1} lim x → 0 1 − cos x x = lim x → 0 cos x − 1 x = 0 {\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x}}=\lim _{x\to 0}{\frac {\cos x-1}{x}}=0} [ 4] [ 8] [ 9] lim x → 0 1 − cos x x 2 = 1 2 {\displaystyle \lim _{x\to 0}{\frac {1-\cos x}{x^{2}}}={\frac {1}{2}}} lim x → n ± tan ( π x + π 2 ) = ∓ ∞ {\displaystyle \lim _{x\to n^{\pm }}\tan \left(\pi x+{\frac {\pi }{2}}\right)=\mp \infty } , for integer n . lim x → 0 tan x x = 1 {\displaystyle \lim _{x\to 0}{\frac {\tan x}{x}}=1} . Or, in general, lim x → 0 tan a x a x = 1 {\displaystyle \lim _{x\to 0}{\frac {\tan ax}{ax}}=1} , for a not equal to 0. lim x → 0 tan a x b x = a b {\displaystyle \lim _{x\to 0}{\frac {\tan ax}{bx}}={\frac {a}{b}}} , for b not equal to 0. lim n → ∞ sin sin ⋯ sin ( x 0 ) ⏟ n = 0 {\displaystyle \lim _{n\to \infty }\ \underbrace {\sin \sin \cdots \sin(x_{0})} _{n}=0} , where x 0 is an arbitrary real number. lim n → ∞ cos cos ⋯ cos ( x 0 ) ⏟ n = d {\displaystyle \lim _{n\to \infty }\ \underbrace {\cos \cos \cdots \cos(x_{0})} _{n}=d} , where d is the Dottie number . x 0 can be any arbitrary real number. In general, any infinite series is the limit of its partial sums . For example, an analytic function is the limit of its Taylor series , within its radius of convergence .
lim n → ∞ ∑ k = 1 n 1 k = ∞ {\displaystyle \lim _{n\to \infty }\sum _{k=1}^{n}{\frac {1}{k}}=\infty } . This is known as the harmonic series .[ 6] lim n → ∞ ( ∑ k = 1 n 1 k − log n ) = γ {\displaystyle \lim _{n\to \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\log n\right)=\gamma } . This is the Euler Mascheroni constant . Notable special limits [ edit ] lim n → ∞ n n ! n = e {\displaystyle \lim _{n\to \infty }{\frac {n}{\sqrt[{n}]{n!}}}=e} lim n → ∞ ( n ! ) 1 / n = ∞ {\displaystyle \lim _{n\to \infty }\left(n!\right)^{1/n}=\infty } . This can be proven by considering the inequality e x ≥ x n n ! {\displaystyle e^{x}\geq {\frac {x^{n}}{n!}}} at x = n {\displaystyle x=n} . lim n → ∞ 2 n 2 − 2 + 2 + ⋯ + 2 ⏟ n = π {\displaystyle \lim _{n\to \infty }\,2^{n}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+\dots +{\sqrt {2}}}}}}}} _{n}=\pi } . This can be derived from Viète's formula for π . Asymptotic equivalences [ edit ] Asymptotic equivalences , f ( x ) ∼ g ( x ) {\displaystyle f(x)\sim g(x)} , are true if lim x → ∞ f ( x ) g ( x ) = 1 {\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=1} . Therefore, they can also be reframed as limits. Some notable asymptotic equivalences include
lim x → ∞ x / ln x π ( x ) = 1 {\displaystyle \lim _{x\to \infty }{\frac {x/\ln x}{\pi (x)}}=1} , due to the prime number theorem , π ( x ) ∼ x ln x {\displaystyle \pi (x)\sim {\frac {x}{\ln x}}} , where π(x) is the prime counting function . lim n → ∞ 2 π n ( n e ) n n ! = 1 {\displaystyle \lim _{n\to \infty }{\frac {{\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}}{n!}}=1} , due to Stirling's approximation , n ! ∼ 2 π n ( n e ) n {\displaystyle n!\sim {\sqrt {2\pi n}}\left({\frac {n}{e}}\right)^{n}} . The behaviour of functions described by Big O notation can also be described by limits. For example
f ( x ) ∈ O ( g ( x ) ) {\displaystyle f(x)\in {\mathcal {O}}(g(x))} if lim sup x → ∞ | f ( x ) | g ( x ) < ∞ {\displaystyle \limsup _{x\to \infty }{\frac {|f(x)|}{g(x)}}<\infty }