# Lim as x->a f(x) = L PROOF using epsilon and delta

• finishimx7
In summary, the limit as x approaches a of f(x) = L is the value that f(x) approaches as x gets closer and closer to a. We can prove the existence of a limit using epsilon and delta, where we must show that for any given epsilon, there exists a corresponding delta that ensures the distance between x and a is less than delta, and the distance between f(x) and L is less than epsilon. Epsilon and delta are significant in formalizing the concept of a limit and providing a rigorous mathematical framework for its existence. If for any epsilon we cannot find a corresponding delta, then the limit does not exist. However, the epsilon-delta proof has limitations and may not work for all functions, requiring other methods
finishimx7
Suppose limx->a f(x) = L does NOT equal 0.

Prove that there exists a (delta) d > 0 such that 0<|x-a|<d

which implies f(x) does NOT equal 0.

Does Anybody Know the Proof For This?

Make epsilon less = |L|.

Last edited:

## 1. What is the definition of the limit as x approaches a of f(x) = L?

The limit as x approaches a of f(x) = L is defined as the value that f(x) approaches as x gets closer and closer to a. This means that no matter how small of an interval around a we choose, there exists a corresponding interval around L that f(x) will always fall within.

## 2. How do we prove that a limit exists using epsilon and delta?

To prove that a limit exists using epsilon and delta, we must show that for any given epsilon (a small positive number), we can find a corresponding delta (a small positive number) such that if the distance between x and a is less than delta, then the distance between f(x) and L is less than epsilon.

## 3. What is the significance of using epsilon and delta in the proof?

Epsilon and delta are used in the proof to formalize the concept of a limit and to provide a rigorous mathematical framework for proving its existence. This method allows us to precisely define what it means for a function to approach a specific value as x gets closer and closer to a.

## 4. Can a limit exist if epsilon and delta cannot be found?

No, if for any epsilon we cannot find a corresponding delta, then the limit does not exist. This means that there is no specific value that f(x) approaches as x approaches a, and the function may be undefined or approach different values from different directions.

## 5. Are there any limitations to using the epsilon-delta proof for limits?

Yes, the epsilon-delta proof only works for functions that are continuous at a and have a well-defined limit. It also requires a lot of mathematical rigor and can be challenging to apply in more complex situations. In these cases, other methods such as the squeeze theorem or L'Hopital's rule may be more useful for proving the existence of a limit.

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