Question

How do you prove the epsilon-delta definition of a limit?

Answer 1

The epsilon-delta definition of a limit is a mathematical approach used to rigorously prove the existence of a limit. It involves using the concepts of epsilon and delta to determine the relationship between the input values and the output values of a function.

Step-by-step explanation:

The epsilon-delta definition of a limit is a mathematical approach used to rigorously prove the existence of a limit. It involves using the concepts of epsilon and delta to determine the relationship between the input values and the output values of a function.

To prove the epsilon-delta definition of a limit, the following steps can be followed:

1. Start with an arbitrary positive value of epsilon, which represents the desired level of accuracy.
2. Find a corresponding positive value of delta that will make the output of the function within epsilon distance of the desired limit value.
3. Show that whenever the input is within delta distance of the limit value, the output is within epsilon distance of the desired limit value.
4. Prove these steps for all possible values of epsilon, which establishes the existence of the limit.