Limit

A limit is the value that a function approaches as the input approaches some value.

Description

The limit of f as x approaches infinity is notated as:

Vector-Values Function

The limit of the vector-values function r(t) = f(t)i + g(t)j + h(t)k as t approaches a is given by:

provided that the component limits exist.

Multivariate Function

One way to express the limit of a multivariate function is to imagine a disk (or ball) around a target point. The disk (ball) is principally formed by all input coordinates no more than δ away from the target coordinate (leaving out the target itself). This set of input coordinates map to a set of outputs characterized by |f(·) - L| < ε, i.e. these outputs are at most ε away from the limit. As δ shrinks to 0, so too does the set of input coordinates, and so too does ε.

Consider the limit of f(x,y) as it approaches (a,b). There is a disk of (x,y) coordinates around (a,b) such that:

The limit is found by shrinking δ. This is notated as:

CategoryRicottone