--- title: "List of limits" chunk: 3/3 source: "https://en.wikipedia.org/wiki/List_of_limits" category: "reference" tags: "science, encyclopedia" date_saved: "2026-05-05T08:15:28.347754+00:00" instance: "kb-cron" --- 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 → ∞ 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 == 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} . 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} 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 x0 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. x0 can be any arbitrary real number. == Sums == 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. 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 == 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. == Limiting behavior == === Asymptotic equivalences === 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}} . === Big O notation === 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 } == References ==