What does it mean for a limit to be unique?

What does it mean for a limit to be unique?

The uniqueness theorem for limits states that if the limit of exists at (in the sense of existence as a finite real number) then it is unique.

How do you prove a function is unique?

Note: To prove uniqueness, we can do one of the following: (i) Assume ∃x, y ∈ S such that P(x) ∧ P(y) is true and show x = y. (ii) Argue by assuming that ∃x, y ∈ S are distinct such that P(x) ∧ P(y), then derive a contradiction. To prove uniqueness and existence, we also need to show that ∃x ∈ S such that P(x) is true.

How do you describe the limit of a function?

The limit of a function at a point a in its domain (if it exists) is the value that the function approaches as its argument approaches. Informally, a function is said to have a limit L at a if it is possible to make the function arbitrarily close to L by choosing values closer and closer to a.

What does the limit of a function tell you?

A limit tells us the value that a function approaches as that function’s inputs get closer and closer to some number. The idea of a limit is the basis of all calculus.

How do you prove a limit is unique?

Theorem 3.1 If a sequence of real numbers {an}n∈N has a limit, then this limit is unique. Proof by contradiction. We hope to prove “For all convergent sequences the limit is unique”.

How would you describe a limit?

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.

Can a sequence have two limits?

Can a sequence have more than one limit? Common sense says no: if there were two different limits L and L′, the an could not be arbitrarily close to both, since L and L′ themselves are at a fixed distance from each other. This is the idea behind the proof of our first theorem about limits.