If you know, the simplest example is a space with at least two points and the indiscrete topology, then every sequence converges to every point. All metric spaces (hence all subsets of an Rn in the usual topology) are Hausdorff, in those spaces, a sequence can have at most one limit..
Keeping this in view, can a sequence converge to two different limits?
A sequence {xn} converges to L if and only if every subsequence of {xn} converges to L. Therefore, if there exists two subsequences {xnk} and {xnl} converging to two different limits L′ and L″, then {xn} cannot be convergent.
Secondly, can a convergent sequence have more than one limit? Hence our assumption must be false, that is, there does not exist a se- quence with more than one limit. Hence for all convergent sequences the limit is unique. Notation Suppose {an}n∈N is convergent. Then by Theorem 3.1 the limit is unique and so we can write it as l, say.
Besides, can a function have more than one limit?
In real function space in talking about limits as inputs approach infinity, no, there are not. In the first case, you have a limit on one point. Otherwise, you don't have a limit. Since you could do this on either positive or negative infinity, you can have up to two limits.
Can sequence converge to zero?
Why some people say it's true: When the terms of a sequence that you're adding up get closer and closer to 0, the sum is converging on some specific finite value. Therefore, as long as the terms get small enough, the sum cannot diverge.
Related Question Answers
How do you know if a sequence converges?
Notice that a sequence converges if the limit as n approaches infinity of An equals a constant number, like 0, 1, pi, or -33. However, if that limit goes to +-infinity, then the sequence is divergent.Why does a sequence have a limit?
The limit of a sequence is the value the sequence approaches as the number of terms goes to infinity. Not every sequence has this behavior: those that do are called convergent, while those that don't are called divergent. Limits capture the long-term behavior of a sequence and are thus very useful in bounding them.Can a finite sequence converge?
Yes. A finite sequence is convergent. It is finite, so it has a last term, say am=M. An sequence converges to a limit L if for any ϵ>0, there exists some integer N such that if k≥N, |ak−L|<ϵ.Does a constant sequence converge?
EXAMPLE 1.3 Every constant sequence is convergent to the constant term in the sequence. To see this, let an = a for all n ∈ N. Then, for every ε > 0, we have |an − a| = 0 < ε ∀ n ≥ N := 1. Then an → 0 as n → ∞.Does every sequence have a limit?
The limit of a sequence is the value the sequence approaches as the number of terms goes to infinity. Not every sequence has this behavior: those that do are called convergent, while those that don't are called divergent. Limits capture the long-term behavior of a sequence and are thus very useful in bounding them.What does it mean when a sequence converges?
When a sequence converges, that means that as you get further and further along the sequence, the terms get closer and closer to a specific limit (usually a real number). A series is a sequence of sums.Are sequences infinite?
This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be finite, as in these examples, or infinite, such as the sequence of all even positive integers (2, 4, 6, ).Can you divide limits?
The multiplication rule for limits says that the product of the limits is the same as the limit of the product of two functions. For example, f(x) = (x - 4)(x - 6)/2(x - 6) is undefined at the value x = 6 because dividing by 2(6 - 6) = 0 is just not feasible.Does limit exist if zero?
In order to say the limit exists, the function has to approach the same value regardless of which direction x comes from (We have referred to this as direction independence). Since that isn't true for this function as x approaches 0, the limit does not exist.What are the rules of limits?
This rule states that the limit of the sum of two functions is equal to the sum of their limits: limx→a[f(x)+g(x)]=limx→af(x)+limx→ag(x).Can you factor constants out of limits?
In other words, we can “factor” a multiplicative constant out of a limit. So, to take the limit of a sum or difference all we need to do is take the limit of the individual parts and then put them back together with the appropriate sign. This is also not limited to two functions.Do limits exist at sharp turns?
In case of a sharp point, the slopes differ from both sides. In the case of a sharp point, the limit from the positive side differs from the limit from the negative side, so there is no limit. The derivative at that point does not exist.Does limit exist if infinity?
exists if and only if it is equal to a number. Note that ∞ is not a number. For example limx→01x2=∞ so it doesn't exist. When a function approaches infinity, the limit technically doesn't exist by the proper definition, that demands it work out to be a number.What is the limit of a number?
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 in general) and are used to define continuity, derivatives, and integrals.What is limit of a constant?
The limit of a sum is equal to the sum of the limits. The limit of a constant function is equal to the constant.Is 1 N convergent or divergent?
n=1 an converge or diverge together. n=1 an converges. n=1 an diverges.What is a convergent sequence give two examples?
Mathwords: Convergent Sequence. A sequence with a limit that is a real number. For example, the sequence 2.1, 2.01, 2.001, 2.0001, . . . has limit 2, so the sequence converges to 2. On the other hand, the sequence 1, 2, 3, 4, 5, 6, . . . has a limit of infinity (∞).What is the limit of convergence?
The limit of a sequence is the value the sequence approaches as the number of terms goes to infinity. Not every sequence has this behavior: those that do are called convergent, while those that don't are called divergent.What is the limit chain rule?
The Chain Rule: What does the chain rule mean? Given a function, f(g(x)), we set the inner function equal to g(x) and find the limit, b, as x approaches a. We then replace g(x) in f(g(x)) with u to get f(u). Using b, we find the limit, L, of f(u) as u approaches b. 
