Limit of a function at an accumulation point

In summary: Therefore, in summary, we have proven that (a) and (b) are equivalent statements for f having a limit at c.
  • #1
monkey372
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Homework Statement


Let f : D → R and c ∈ R an accumulation point of D.

Homework Equations



Prove the following are equivalent:[/b]

(a) f has a limit at c.

(b) For all sequences (sn ) such that c = sn ∈ D for all n ∈ N and sn → c,
the sequence (f (sn )) is convergent in R.


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.

But then I don't know how I should continue. Please help me out.
 
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  • #2


Hello! Your approach is a good start. Here's how you can continue:

Since f has a limit at c, we know that for any ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε.

Now, for any sequence (sn) such that c = sn ∈ D for all n ∈ N and sn → c, we know that there exists a positive integer N such that for all n > N, 0 < |sn - c| < δ. This means that for all n > N, |f(sn) - L| < ε.

Since this is true for any ε > 0, we can say that the sequence (f(sn)) converges to L. Therefore, (a) implies (b).

To show that (b) implies (a), we can use a proof by contradiction. Assume that (a) is false, i.e. f does not have a limit at c. This means that there exists an ε > 0 such that for any δ > 0, there exists a point x in D such that 0 < |x - c| < δ but |f(x) - L| ≥ ε, where L is the supposed limit of f at c.

Now, consider the sequence (xn) defined as follows: xn = c + (1/n). Note that xn ∈ D for all n ∈ N and xn → c. However, for this sequence, we have |f(xn) - L| ≥ ε for all n ∈ N, which contradicts the fact that (f(xn)) converges to L. Therefore, our assumption that (a) is false must be incorrect, and hence (b) implies (a).

Hence, we have shown that (a) and (b) are equivalent.
 

FAQ: Limit of a function at an accumulation point

What is the definition of 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 approaches the accumulation point from both sides. It is a fundamental concept in calculus that is used to describe the behavior of a function near a specific point.

How is the limit of a function at an accumulation point different from the limit at a regular point?

The limit of a function at an accumulation point is defined as the limit from both sides, while the limit at a regular point is defined as the limit from one side. This means that the limit at an accumulation point takes into account the behavior of the function from both directions, while the limit at a regular point only considers one direction.

What is the importance of the limit of a function at an accumulation point?

The limit of a function at an accumulation point is important because it helps us understand the behavior of a function near a specific point. It allows us to determine if a function is continuous at that point, and also helps us to find the derivative and integral of a function at that point.

How do you find the limit of a function at an accumulation point?

To find the limit of a function at an accumulation point, you must first determine the behavior of the function as the input approaches the accumulation point from both sides. This can be done by plugging in values that are close to the accumulation point and observing the output. If the function approaches the same value from both directions, then that value is the limit at the accumulation point.

Can the limit of a function at an accumulation point exist if the function is not defined at that point?

Yes, the limit of a function at an accumulation point can exist even if the function is not defined at that point. This is because the limit is based on the behavior of the function near the accumulation point, not the value at that point. As long as the function approaches the same value from both sides, the limit exists.

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