Understanding the Epsilon-Delta Definition of Limit in Calculus

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In summary, the conversation discusses the epsilon-delta definition of limit and the delta neighborhood concept. The definition states that if x approaches p, then f(x) will approach y. To verify this, one must ensure that f gets closer to y as x gets closer to p. This can be done by specifying a small interval around y and finding an interval around p of a certain length where f maps within that deviation, regardless of how small the interval is.
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indra00
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please guys! some one please help me understand the epsilon-delta definition of limit and the delta nbd concept.
 
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What have you been taught so far?
 
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It's actually very simple. What does it intuitively mean when we say that [tex]\lim_{x\to p}f(x)=y[/tex]? It means that if you let x approach p then f(x) will approach y. How to see if that is really the case? Well, if the function f is really going to get closer and closer to y if you let x appoach p closer and closer, then it should be the case that if you specify some small interval [tex]\delta[/tex] containing y you can find an interval around p of length [tex]\epsilon[/tex] such that the function f will map that interval within that deviation, no matter how small you make [tex]\delta[/tex].
 

FAQ: Understanding the Epsilon-Delta Definition of Limit in Calculus

1) What is a limit in mathematics?

A limit in mathematics is the value that a function approaches as the input value gets closer and closer to a given point. It represents the behavior of the function near a specific point.

2) Why is understanding limits important?

Understanding limits is important because it is a fundamental concept in calculus and is used to solve various mathematical problems. It also helps in understanding the behavior of a function and its properties.

3) How do you find the limit of a function?

To find the limit of a function, you can use various methods such as substitution, factoring, rationalization, and L'Hôpital's rule. It is also important to understand the behavior of the function and the properties of limits.

4) What are the common types of limits?

The common types of limits include one-sided limits, infinite limits, and limits at infinity. One-sided limits are used when the function approaches from one direction, while infinite limits are used when the function approaches either positive or negative infinity. Limits at infinity are used when the function approaches infinity as the input value grows without bound.

5) How can understanding limits help in real-life applications?

Understanding limits can help in real-life applications such as predicting the growth of a population, calculating the speed of an object, and determining the maximum and minimum values of a function. It is also used in various fields such as physics, engineering, and economics.

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