Limit

Limit is a fundamental concept in mathematics, particularly in the fields of calculus and analysis. It describes the behavior of a function as its argument approaches a certain point, or as the terms of a sequence approach a certain value. Limits are essential for defining derivatives, integrals, and continuity.

Definition

In mathematics, the limit of a function is a fundamental concept that describes the behavior of that function as its input approaches a given point. The formal definition, known as the ε-δ definition of a limit, was first introduced in the 19th century by Augustin-Louis Cauchy and Karl Weierstrass.

Limit of a Function

The limit of a function f(x) as x approaches a value a is denoted as:

\(\lim_Template:X \to a f(x) = L\)

This notation means that as x gets closer and closer to a, the function values f(x) get arbitrarily close to L.

Limit of a Sequence

Similarly, the limit of a sequence \((a_n)\) is the value that the sequence approaches as the number of terms goes to infinity. It is denoted as:

\(\lim_Template:N \to \infty a_n = L\)

This means that for any small positive number ε, there exists a natural number N such that for all n greater than N, the absolute difference between a_n and L is less than ε.

Types of Limits

There are several types of limits in mathematical analysis:

• One-sided limits: Limits taken from only one side (left or right) of the point of interest.
• Infinite limits: Limits in which the function approaches infinity as the input approaches a certain value.
• Limit at infinity: Describes the behavior of a function as the input grows without bound.

Applications

Limits are used in various areas of mathematics and applied sciences:

• In calculus, limits are used to define the derivative and the integral.
• In real analysis, limits underpin the definitions of continuity, metric spaces, and topological spaces.
• In physics, limits describe the behavior of physical quantities under extreme conditions.

See Also

• Derivative
• Integral
• Continuity (mathematics)
• L'Hôpital's rule