Limit

In mathematics, a limit is a value approached by a function or sequence as its input approaches a specified point or its index increases without bound. Limits make the idea of “arbitrarily close” precise without requiring that the limiting value ever be attained. They underpin calculus and mathematical analysis, providing definitions of continuity, differentiation, integration, and infinite summation. (openstax.org)

Limits of functions

For a real-valued function $f$, the notation

means that $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close to, but different from, $a$. Formally, the epsilon–delta definition requires that for every $\varepsilon>0$, there exists $\delta>0$ such that

for every $x$ in the function’s domain. The point $a$ must be an accumulation point of that domain. The order of the quantifiers matters: $\delta$ may depend on the chosen tolerance $\varepsilon$, but must work for all qualifying inputs. (openstax.org)

A limit depends on nearby values, not on the value at the point itself. For example,

because the quotient equals $x+1$ whenever $x\ne1$. The original expression is undefined at $1$; assigning any value there leaves the limit unchanged. By contrast, a continuous function at $a$ satisfies $\lim_{x\to a}f(x)=f(a)$, with continuity understood relative to its domain. (openstax.org)

One-sided and infinite limits

A one-sided limit restricts the direction of approach. The notations $x\to a^-$ and $x\to a^+$ indicate approach from below and above. When both sides are available in the domain, a finite two-sided limit exists exactly when the two one-sided limits exist and agree. Thus $|x|/x$ has left-hand limit $-1$ and right-hand limit $1$ at zero, but no two-sided limit. (openstax.org)

Limits also describe behavior as inputs increase without bound. For example,

An infinite limit, such as $\lim_{x\to0}1/x^2=+\infty$, instead expresses growth beyond every prescribed positive bound. Infinity here is not an ordinary real number or a finite limiting value. A finite limit at infinity describes a horizontal asymptote, whereas an infinite limit at a finite point indicates a vertical asymptote. (openstax.org)

Sequences and series

A sequence $(a_n)$ converges to $L$ if, for every $\varepsilon>0$, there exists a positive integer $N$ such that

Consequently, every sufficiently late term—not merely selected terms—must lie within the tolerance. The sequence $1/n$ converges to zero, while $(-1)^n$ does not converge: its even and odd subsequences approach different values. A convergent real sequence has a unique limit and is bounded, although boundedness alone does not ensure convergence. (openstax.org)

An infinite series is defined through its partial sums:

provided this limit exists and is finite. Terms approaching zero are necessary but insufficient for convergence; the harmonic series $\sum 1/n$ diverges. A Cauchy sequence instead requires sufficiently late terms to be arbitrarily close to one another. Every such sequence of real numbers converges, reflecting the completeness of the real number system. (ocw.mit.edu)

Limit laws and calculation

Finite limits obey algebraic laws: limits of sums, differences, and products equal the corresponding operations on their limits. The quotient law applies when the denominator’s limit is nonzero. These rules explain why a polynomial can be evaluated by direct substitution, and why a rational function permits substitution wherever its denominator is nonzero. (openstax.org)

An expression such as $0/0$ is an indeterminate form, not a limit value. Different expressions with this substitution pattern can have different limits. Factoring, rationalizing, or estimating may reveal the relevant behavior. The squeeze theorem establishes a limit when a function is trapped between two functions approaching the same value. Numerical tables and graphs can suggest a result, but do not replace a mathematical proof. (openstax.org)

Roles in calculus and broader analysis

The derivative is a limit of difference quotients:

It measures instantaneous change when this finite limit exists. A definite integral in the Riemann sense is the limit of Riemann sums as the largest subinterval width tends to zero, with the same limiting value required independently of the sample points. (openstax.org)

In a metric space, distance replaces absolute value in the definition of convergence. Topology generalizes the idea further through neighborhoods. For sequences of functions, pointwise convergence allows convergence to be checked separately at each input, whereas uniform convergence requires one index threshold to control the error throughout the domain. This distinction determines which properties survive passage to a limit: a uniform limit of continuous functions is continuous, but a pointwise limit need not be. Interchanging limits with differentiation or integration requires additional hypotheses. (math.colostate.edu)