Limit

A limit describes the value that a function or sequence approaches as the input or index approaches some value.

Limit of a Sequence

A sequence $(a_n)$ has limit $L$ if for every $\epsilon > 0$, there exists $N \in \mathbb{N}$ such that $n \geq N$ implies $|a_n - L| < \epsilon$.

$$ \lim_{n \to \infty} a_n = L $$

See convergence for more details.

Limit of a Function

A function $f$ has limit $L$ as $x$ approaches $c$ if for every $\epsilon > 0$, there exists $\delta > 0$ such that $0 < |x - c| < \delta$ implies $|f(x) - L| < \epsilon$.

$$ \lim_{x \to c} f(x) = L $$

See functional_limit for more details.

Properties

Limits satisfy the Algebraic_Limit_Theorem:

• $\lim(a_n + b_n) = \lim a_n + \lim b_n$
• $\lim(c \cdot a_n) = c \cdot \lim a_n$
• $\lim(a_n \cdot b_n) = \lim a_n \cdot \lim b_n$

Related

• convergence
• functional_limit
• sequence
• continuity