# Formal definition of limits Part 4: using the definition | AP Calculus AB | Khan Academy | Summary and Q&A

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January 12, 2013
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Formal definition of limits Part 4: using the definition | AP Calculus AB | Khan Academy

## TL;DR

The video explains the epsilon-delta definition of limits and demonstrates how to use it to rigorously prove the existence of a limit.

## Questions & Answers

### Q: What does the epsilon-delta definition of limits state?

The epsilon-delta definition states that for a limit to exist, there must be a positive epsilon that determines how close the function's output needs to be to the limit, and a corresponding delta that determines the distance from the input to the limit.

### Q: How is the concept of limits applied in the video?

The video applies the concept of limits to prove the existence of the limit of a function as x approaches a specific value.

### Q: How is the example function defined in the video?

The example function, f(x), is defined to be equal to 2x for all values of x except when x is 5. For x = 5, f(x) is equal to 5 itself.

### Q: What is the goal of the video?

The goal of the video is to demonstrate how to use the epsilon-delta definition to rigorously prove the existence of the limit of the example function as x approaches 5.

## Summary & Key Takeaways

• The epsilon-delta definition of limits states that for a limit to exist, there must be a positive epsilon that determines how close the function's output needs to be to the limit, and a corresponding delta that determines the distance from the input to the limit.

• The video presents an example function, where f(x) is equal to 2x for all x except when x is 5, in which case it is equal to x.

• Using the epsilon-delta definition, the video aims to prove that the limit of f(x) as x approaches 5 is equal to 10.