# Anyone can help me out? Limit of a function at an accumulation point.

• monkey372
In summary: Therefore, we have proven that (a) and (b) are equivalent. In summary, the problem asks to prove that the following statements are equivalent: (a) f has a limit at c, and (b) For all sequences (sn ) such that c does not equal to sn ∈ D for all n ∈ N and sn → c, the sequence (f (sn )) is convergent in R. This is proven by showing that (a) implies (b) and (b) implies (a) through the use of limit definitions and the convergence of sequences.
monkey372
1. Homework Statement
Let f : D → R and c ∈ R an accumulation point of D.

2. Homework Equations

Prove the following are equivalent:

(a) f has a limit at c.

(b) For all sequences (sn ) such that c does not equal to sn ∈ D for all n ∈ N and sn → c,
the sequence (f (sn )) is convergent in R.3. The Attempt at a Solution
Here is my approach:
Let x be an arbitrary element of D. Since f has a limit at c which means f(x) -> L as x -> c, we have the sequence xn -> c, where (xn) does not equal to c for all natural number n, as f(xn)->c.

## The Attempt at a Solution

Last edited:
We will prove (a) implies (b). Suppose f has a limit at c, then for all sequences (sn) such that c does not equal to sn ∈ D for all n ∈ N and sn → c, we have that lim f(sn) = lim f(x) as x -> c. Since the sequence (sn) converges to c, it follows that the sequence (f(sn)) converges to the same limit, namely f(c). Thus, (b) is true.The proof of (b) implies (a) follows similarly. If for all sequences (sn) such that c does not equal to sn ∈ D for all n ∈ N and sn → c, we have that the sequence (f(sn)) converges to some limit L, then it follows that f(x) -> L as x -> c. Thus, (a) is true.

## 1. What is a limit of a function at an accumulation point?

The limit of a function at an accumulation point is the value that the function approaches as the input values get closer and closer to the accumulation point. It is an important concept in calculus and is used to determine the behavior of a function at a particular point.

## 2. How is the limit of a function at an accumulation point calculated?

The limit of a function at an accumulation point is calculated by evaluating the function at values that are closer and closer to the accumulation point. This can be done algebraically or graphically, and the resulting value is the limit.

## 3. What does it mean if the limit of a function at an accumulation point does not exist?

If the limit of a function at an accumulation point does not exist, it means that the function does not approach a specific value as the input values get closer and closer to the accumulation point. This can happen if the function has a discontinuity or if it oscillates between multiple values.

## 4. Can the limit of a function at an accumulation point be different from the value of the function at that point?

Yes, the limit of a function at an accumulation point can be different from the value of the function at that point. This can occur if the function is not continuous at the accumulation point or if there is a jump or hole at that point.

## 5. How is the limit of a function at an accumulation point related to continuity?

The limit of a function at an accumulation point is related to continuity because a function is continuous at a point if and only if the limit of the function at that point exists and is equal to the value of the function at that point. In other words, for a function to be continuous, the limit and the value must be the same at every point.

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