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This is alist oflimits for commonfunctions such aselementary functions. In this article, the termsa,b andc are constants with respect tox.

${\displaystyle \underset{x\to c}{lim}f(x)=L}$if and only if. This is the(ε, δ)-definition of limit.

Thelimit superior and limit inferior of a sequence are defined as${\displaystyle \underset{n\to \mathrm{\infty }}{lim sup}{x}_n=\underset{n\to \mathrm{\infty }}{lim}\left(\underset{m\ge n}{sup}{x}_m\right)}$ and${\displaystyle \underset{n\to \mathrm{\infty }}{lim inf}{x}_n=\underset{n\to \mathrm{\infty }}{lim}\left(\underset{m\ge n}{inf}{x}_m\right)}$.

A function,${\displaystyle f(x)}$, is said to be continuous at a point,c, if$${\displaystyle \underset{x\to c}{lim}f(x)=f(c).}$$

If${\displaystyle \underset{x\to c}{lim}f(x)=L}$ then:

• ${\displaystyle \underset{x\to c}{lim}[f(x)\pm a]=L\pm a}$
• ${\displaystyle \underset{x\to c}{lim}af(x)=aL}$^[1][2][3]
• ${\displaystyle \underset{x\to c}{lim}\frac{1}{f(x)}=\frac{1}{L}}$^[4] if L is not equal to 0.
• ${\displaystyle \underset{x\to c}{lim}f(x{)}^n=L^n}$ ifn is a positive integer^[1][2][3]
• ${\displaystyle \underset{x\to c}{lim}f(x{)}^{\frac{1}{n}}=L^{\frac{1}{n}}}$ ifn is a positive integer, and ifn is even, thenL > 0.^[1][3]

In general, ifg(x) is continuous atL and${\displaystyle \underset{x\to c}{lim}f(x)=L}$ then

• ${\displaystyle \underset{x\to c}{lim}g\left(f(x)\right)=g(L)}$^[1][2]

If${\displaystyle \underset{x\to c}{lim}f(x)=L_1}$ and${\displaystyle \underset{x\to c}{lim}g(x)=L_2}$ then:

• ${\displaystyle \underset{x\to c}{lim}[f(x)\pm g(x)]=L_1\pm L_2}$^[1][2][3]
• ${\displaystyle \underset{x\to c}{lim}[f(x)g(x)]=L_1\cdot L_2}$^[1][2][3]
• ${\displaystyle \underset{x\to c}{lim}\frac{f(x)}{g(x)}=\frac{L_1}{L_2}\text{ if }{L}_2\ne 0}$^[1][2][3]

In these limits, the infinitesimal change${\displaystyle h}$ is often denoted${\displaystyle \mathrm{\Delta }x}$ or${\displaystyle \delta x}$. If${\displaystyle f(x)}$ isdifferentiable at${\displaystyle x}$,

• ${\displaystyle \underset{h\to 0}{lim}\frac{f(x+h)-f(x)}{h}=f^{\prime }(x)}$. This is the definition of thederivative. Alldifferentiation rules can also be reframed as rules involving limits. For example, ifg(x) is differentiable atx,
  • ${\displaystyle \underset{h\to 0}{lim}\frac{f\circ g(x+h)-f\circ g(x)}{h}=f^{\prime }[g(x)]g^{\prime }(x)}$. This is thechain rule.
  • ${\displaystyle \underset{h\to 0}{lim}\frac{f(x+h)g(x+h)-f(x)g(x)}{h}=f^{\prime }(x)g(x)+f(x)g^{\prime }(x)}$. This is theproduct rule.
• ${\displaystyle \underset{h\to 0}{lim}{\left(\frac{f(x+h)}{f(x)}\right)}^{1/h}=\exp \left(\frac{f^{\prime }(x)}{f(x)}\right)}$
• ${\displaystyle \underset{h\to 0}{lim}{\left(\frac{f(e^{h}x)}{f(x)}\right)}^{1/h}=\exp \left(\frac{x{f}^{\prime }(x)}{f(x)}\right)}$

If${\displaystyle f(x)}$ and${\displaystyle g(x)}$ are differentiable on an open interval containingc, except possiblyc itself, and${\displaystyle \underset{x\to c}{lim}f(x)=\underset{x\to c}{lim}g(x)=0\text{ or }\pm \mathrm{\infty }}$,L'Hôpital's rule can be used:

• ${\displaystyle \underset{x\to c}{lim}\frac{f(x)}{g(x)}=\underset{x\to c}{lim}\frac{f^{\prime }(x)}{g^{\prime }(x)}}$^[2]

If${\displaystyle f(x)\le g(x)}$ for all x in an interval that containsc, except possiblyc itself, and the limit of${\displaystyle f(x)}$ and${\displaystyle g(x)}$ both exist atc, then^[5]$${\displaystyle \underset{x\to c}{lim}f(x)\le \underset{x\to c}{lim}g(x)}$$

If${\displaystyle \underset{x\to c}{lim}f(x)=\underset{x\to c}{lim}h(x)=L}$ and${\displaystyle f(x)\le g(x)\le h(x)}$ for allx in anopen interval that containsc, except possiblyc itself,$${\displaystyle \underset{x\to c}{lim}g(x)=L.}$$ This is known as thesqueeze theorem.^[1][2] This applies even in the cases thatf(x) andg(x) take on different values atc, or are discontinuous atc.

• ${\displaystyle \underset{x\to c}{lim}a=a}$^[1][2][3]

• ${\displaystyle \underset{x\to c}{lim}x=c}$^[1][2][3]
• ${\displaystyle \underset{x\to c}{lim}(ax+b)=ac+b}$
• ${\displaystyle \underset{x\to c}{lim}{x}^n=c^n}$ ifn is a positive integer^[5]
• 

In general, if${\displaystyle p(x)}$ is a polynomial then, by the continuity of polynomials,^[5]$${\displaystyle \underset{x\to c}{lim}p(x)=p(c)}$$ This is also true forrational functions, as they are continuous on theirdomains.^[5]

• ${\displaystyle \underset{x\to c}{lim}{x}^a=c^a.}$^[5] In particular,
  • 
• ${\displaystyle \underset{x\to c}{lim}{x}^{1/a}=c^{1/a}}$.^[5] In particular,
  • ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}{x}^{1/a}=\underset{x\to \mathrm{\infty }}{lim}\sqrt[a]{x}=\mathrm{\infty }\text{ for any }a>0}$^[6]
• ${\displaystyle \underset{x\to 0^{+}}{lim}{x}^{-n}=\underset{x\to 0^{+}}{lim}\frac{1}{x^n}=+\mathrm{\infty }}$
• 
• ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}a{x}^{-1}=\underset{x\to \mathrm{\infty }}{lim}a/x=0\text{ for any real }a}$

• ${\displaystyle \underset{x\to c}{lim}{e}^x=e^c}$, due to the continuity of${\displaystyle e^x}$
• 
• ^[6]
• 

• ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\sqrt[x]{x}=\underset{x\to \mathrm{\infty }}{lim}{x}^{1/x}=1}$

• ${\displaystyle \underset{x\to +\mathrm{\infty }}{lim}{\left(\frac{x}{x+k}\right)}^x=e^{-k}}$^[2]
• ${\displaystyle \underset{x\to 0}{lim}{\left(1+x\right)}^{\frac{1}{x}}=e}$^[2]
• ${\displaystyle \underset{x\to 0}{lim}{\left(1+kx\right)}^{\frac{m}{x}}=e^{mk}}$
• ${\displaystyle \underset{x\to +\mathrm{\infty }}{lim}{\left(1+\frac{1}{x}\right)}^x=e}$^[7]
• ${\displaystyle \underset{x\to +\mathrm{\infty }}{lim}{\left(1-\frac{1}{x}\right)}^x=\frac{1}{e}}$
• ${\displaystyle \underset{x\to +\mathrm{\infty }}{lim}{\left(1+\frac{k}{x}\right)}^{mx}=e^{mk}}$^[6]
• ${\displaystyle \underset{x\to 0}{lim}{\left(1+a\left(e^{-x}-1\right)\right)}^{-\frac{1}{x}}=e^a}$. This limit can be derived fromthis limit.

• ${\displaystyle \underset{x\to 0}{lim}x{e}^{-x}=0}$
• ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}x{e}^{-x}=0}$
• ${\displaystyle \underset{x\to 0}{lim}\left(\frac{a^x-1}{x}\right)=\ln a,}$ for all positivea.^[4][7]
• ${\displaystyle \underset{x\to 0}{lim}\left(\frac{e^x-1}{x}\right)=1}$
• ${\displaystyle \underset{x\to 0}{lim}\left(\frac{e^{ax}-1}{x}\right)=a}$

• ${\displaystyle \underset{x\to c}{lim}\ln x=\ln c}$, due to the continuity of${\displaystyle \ln x}$. In particular,
  • ${\displaystyle \underset{x\to 0^{+}}{lim}\log x=-\mathrm{\infty }}$
  • ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\log x=\mathrm{\infty }}$
• ${\displaystyle \underset{x\to 1}{lim}\frac{\ln (x)}{x-1}=1}$
• ${\displaystyle \underset{x\to 0}{lim}\frac{\ln (x+1)}{x}=1}$^[7]
• ${\displaystyle \underset{x\to 0}{lim}\frac{-\ln \left(1+a\left(e^{-x}-1\right)\right)}{x}=a}$. This limit follows fromL'Hôpital's rule.
• ${\displaystyle \underset{x\to 0}{lim}x\ln x=0}$, hence${\displaystyle \underset{x\to 0}{lim}{x}^x=1}$
• ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{\ln x}{x}=0}$^[6]

Forb > 1,

• ${\displaystyle \underset{x\to 0^{+}}{lim}{\log }_{b}x=-\mathrm{\infty }}$
• ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}{\log }_{b}x=\mathrm{\infty }}$

Forb < 1,

• ${\displaystyle \underset{x\to 0^{+}}{lim}{\log }_{b}x=\mathrm{\infty }}$
• ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}{\log }_{b}x=-\mathrm{\infty }}$

Both cases can be generalized to:

• ${\displaystyle \underset{x\to 0^{+}}{lim}{\log }_{b}x=-F(b)\mathrm{\infty }}$
• ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}{\log }_{b}x=F(b)\mathrm{\infty }}$

where${\displaystyle F(x)=2H(x-1)-1}$ and${\displaystyle H(x)}$ is theHeaviside step function

If${\displaystyle x}$ is expressed in radians:

• ${\displaystyle \underset{x\to a}{lim}\sin x=\sin a}$
• ${\displaystyle \underset{x\to a}{lim}\cos x=\cos a}$

These limits both follow from the continuity of sin and cos.

• ${\displaystyle \underset{x\to 0}{lim}\frac{\sin x}{x}=1}$.^[7][8] Or, in general,
  • ${\displaystyle \underset{x\to 0}{lim}\frac{\sin ax}{ax}=1}$, fora not equal to 0.
  • ${\displaystyle \underset{x\to 0}{lim}\frac{\sin ax}{x}=a}$
  • ${\displaystyle \underset{x\to 0}{lim}\frac{\sin ax}{bx}=\frac{a}{b}}$, forb not equal to 0.
• ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}x\sin \left(\frac{1}{x}\right)=1}$
• ${\displaystyle \underset{x\to 0}{lim}\frac{1-\cos x}{x}=\underset{x\to 0}{lim}\frac{\cos x-1}{x}=0}$^[4][8][9]
• ${\displaystyle \underset{x\to 0}{lim}\frac{1-\cos x}{x^2}=\frac{1}{2}}$
• ${\displaystyle \underset{x\to n^{\pm }}{lim}\tan \left(\pi x+\frac{\pi }{2}\right)=\mp \mathrm{\infty }}$, for integern.
• ${\displaystyle \underset{x\to 0}{lim}\frac{\tan x}{x}=1}$. Or, in general,
  • ${\displaystyle \underset{x\to 0}{lim}\frac{\tan ax}{ax}=1}$, fora not equal to 0.
  • ${\displaystyle \underset{x\to 0}{lim}\frac{\tan ax}{bx}=\frac{a}{b}}$, forb not equal to 0.
• ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\underset{n}{\underbrace{\sin \sin \cdots \sin (x_0)}}=0}$, wherex₀ is an arbitrary real number.
• ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\underset{n}{\underbrace{\cos \cos \cdots \cos (x_0)}}=d}$, where d is theDottie number.x₀ can be any arbitrary real number.

In general, anyinfinite series is the limit of itspartial sums. For example, ananalytic function is the limit of itsTaylor series, within itsradius of convergence.

• ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\sum _{k=1}^n\frac{1}{k}=\mathrm{\infty }}$. This is known as theharmonic series.^[6]
• ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\left(\sum _{k=1}^n\frac{1}{k}-\log n\right)=\gamma }$. This is theEuler Mascheroni constant.

• ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{n}{\sqrt[n]{n!}}=e}$
• ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{\left(n!\right)}^{1/n}=\mathrm{\infty }}$. This can be proven by considering the inequality${\displaystyle e^x\ge \frac{x^n}{n!}}$ at${\displaystyle x=n}$.
• ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}{2}^n\underset{n}{\underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}}=\pi }$. This can be derived fromViète's formula forπ.

Asymptotic equivalences,${\displaystyle f(x)\sim g(x)}$, are true if${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{f(x)}{g(x)}=1}$. Therefore, they can also be reframed as limits. Some notable asymptotic equivalences include

• ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{x/\ln x}{\pi (x)}=1}$, due to theprime number theorem,${\displaystyle \pi (x)\sim \frac{x}{\ln x}}$, where π(x) is theprime counting function.
• ${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\frac{\sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n}{n!}=1}$, due toStirling's approximation,${\displaystyle n!\sim \sqrt{2\pi n}{\left(\frac{n}{e}\right)}^n}$.

The behaviour of functions described byBig O notation can also be described by limits. For example

• ${\displaystyle f(x)\in \mathcal{O}(g(x))}$ if

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