Let (an)(a_n) be a sequence that converges to L∈R.L \in \mathbb{R}. The limit of (an)(a_n) is the number L.L.

L=lim⁡n→∞anL = \lim_{n\rightarrow \infty} a_n

The statement “(an)(a_n) converges to LL” is also written an→La_n \rightarrow L as n→∞.n \rightarrow \infty .

Let (an)(a_n) and (bn)(b_n) be convergent sequences and c∈R.c \in \mathbb{R}. The limit laws state that sums, differences, scalar multiples, products and quotients of convergent sequences converge to the corresponding combinations of their limits.

lim⁡n→∞(an±bn)=lim⁡n→∞an±lim⁡n→∞bn\lim_{n\rightarrow \infty} \left( a_n \pm b_n \right) = \lim_{n\rightarrow \infty} a_n \pm \lim_{n\rightarrow \infty} b_n lim⁡n→∞(c⋅an)=c⋅lim⁡n→∞an\lim_{n\rightarrow \infty} \left( c \cdot a_n \right) = c \cdot \lim_{n\rightarrow \infty} a_n lim⁡n→∞(an⋅bn)=lim⁡n→∞an⋅lim⁡n→∞bn\lim_{n\rightarrow \infty} \left( a_n \cdot b_n \right) = \lim_{n\rightarrow \infty} a_n \cdot \lim_{n\rightarrow \infty} b_n lim⁡n→∞(anbn)=lim⁡n→∞anlim⁡n→∞bn  if  bn≠0 ∀n  and  lim⁡n→∞bn≠0\lim_{n\rightarrow \infty} \left( \frac{a_n}{b_n} \right) = \frac{\lim_{n\rightarrow \infty} a_n}{\lim_{n\rightarrow \infty} b_n} \,\, \mathrm{if} \,\, b_n \neq 0 \, \forall n \,\, \mathrm{and} \,\, \lim_{n\rightarrow \infty} b_n \neq 0

The convergence assumption is necessary: if an=na_n = n and bn=−nb_n = -n then an+bn→0a_n + b_n \rightarrow 0 even though neither lim⁡n→∞an\lim_{n\rightarrow \infty} a_n nor lim⁡n→∞bn\lim_{n\rightarrow \infty} b_n exists.

The following common limits hold:

lim⁡n→∞1n=0\lim_{n\rightarrow \infty} \frac{1}{n} = 0 lim⁡n→∞c=c  ∀c∈R\lim_{n\rightarrow \infty} c = c \, \, \forall c \in \mathbb{R} lim⁡n→∞xn=0  ∀ ∣x∣<1\lim_{n\rightarrow \infty} x^n = 0 \, \, \forall \, \lvert x\rvert < 1 lim⁡n→∞x1n=1  ∀ x>0\lim_{n\rightarrow \infty} x^{\frac{1}{n}} = 1 \, \, \forall \, x > 0

Let X⊆RX \subseteq \mathbb{R} and x0∈R.x_0 \in \mathbb{R}. The point x0x_0 is a limit point of XX if and only if every open interval containing x0x_0 contains a point of XX other than x0.x_0.

Let X⊆R,X \subseteq \mathbb{R}, f:X→Rf: X \rightarrow \mathbb{R} be a function, x0∈Rx_0 \in \mathbb{R} be a limit point of XX and L∈R.L \in \mathbb{R}. The function ff has the limit LL as x→x0x \rightarrow x_0 if and only if for every ε>0\varepsilon > 0 there exists δ>0\delta > 0 such that f(x)f(x) is ε\varepsilon-close to LL for all x∈Xx \in X with 0<∣x−x0∣<δ.0 < \lvert x-x_0\rvert < \delta.

lim⁡x→x0f(x)=L⇔∀ε>0 ∃δ>0:∣f(x)−L∣≤ε ∀x∈X, 0<∣x−x0∣<δ\lim_{x\rightarrow x_0} f(x) = L \Leftrightarrow \forall \varepsilon > 0 \, \exists \delta > 0 : \lvert f(x)-L \rvert \leq \varepsilon \, \forall x \in X, \, 0 < \lvert x-x_0\rvert < \delta

The value f(x0),f(x_0), if it is defined, plays no part in the limit. The limit laws hold for limits of functions in the same form as for limits of sequences.

Let X⊆R,X \subseteq \mathbb{R}, f:X→Rf: X \rightarrow \mathbb{R} be a function and x0∈Rx_0 \in \mathbb{R} be a limit point of X.X. The function ff has the limit LL as x→x0x \rightarrow x_0 if and only if f(xn)→Lf(x_n) \rightarrow L for every sequence (xn)(x_n) of points of X∖{x0}X \setminus \{x_0\} with xn→x0.x_n \rightarrow x_0.