Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Limit Definition and Continuity

In course

MATH-105(a): Advanced analysis II - vector analysis

Etudier les concepts fondamentaux d'analyse et le calcul différentiel et intégral des fonctions réelles de plusieurs variables. ...

Description

This lecture covers the limit definition of a function, including the existence and equality of limits, limit characterization, and demonstrations related to limits. It also discusses continuity on a set, emphasizing the concept of continuity in various contexts.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructors (2)

Boris Buffoni

Fabio Nobile

Official source

Related concepts (9)

Limit of a function

Although the function \tfrac{\sin x}{x} is not defined at zero, as x becomes closer and closer to zero, \tfrac{\sin x}{x} becomes arbitrarily close to 1. In other words, the limit of \tfrac{\sin x}{x}, as x approaches zero, equals 1. In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input. Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x.

Limit (mathematics)

In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to and direct limit in . In formulas, a limit of a function is usually written as (although a few authors use "Lt" instead of "lim") and is read as "the limit of f of x as x approaches c equals L".

Limit inferior and limit superior

In mathematics, the limit inferior and limit superior of a sequence can be thought of as limiting (that is, eventual and extreme) bounds on the sequence. They can be thought of in a similar fashion for a function (see limit of a function). For a set, they are the infimum and supremum of the set's limit points, respectively. In general, when there are multiple objects around which a sequence, function, or set accumulates, the inferior and superior limits extract the smallest and largest of them; the type of object and the measure of size is context-dependent, but the notion of extreme limits is invariant.

Limit of a sequence

As the positive integer becomes larger and larger, the value becomes arbitrarily close to . We say that "the limit of the sequence equals ." In mathematics, the limit of a sequence is the value that the terms of a sequence "tend to", and is often denoted using the symbol (e.g., ). If such a limit exists, the sequence is called convergent. A sequence that does not converge is said to be divergent. The limit of a sequence is said to be the fundamental notion on which the whole of mathematical analysis ultimately rests.

One-sided limit

In calculus, a one-sided limit refers to either one of the two limits of a function of a real variable as approaches a specified point either from the left or from the right. The limit as decreases in value approaching ( approaches "from the right" or "from above") can be denoted: The limit as increases in value approaching ( approaches "from the left" or "from below") can be denoted: If the limit of as approaches exists then the limits from the left and from the right both exist and are equal.

Related lectures (33)

Limits and Continuity

Covers the concept of limits and continuity in real analysis.

Limits of Functions: Characterization and Operations

Explores limit characterization using sequences and algebraic operations on limits.

Limit of a Sequence

Explores the limit of a sequence and its convergence properties, including boundedness and monotonicity.

Generalized Integrals: Convergence and Divergence

Explores the convergence and divergence of generalized integrals using comparison methods and variable transformations.

Limit and Function Composition MOOC: Analysis I MOOC: Analysis I (part 4) : Limit of a function, continuous functions