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epsilon-delta limit definition youtube playlist

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The epsilon-delta definition of the limit is the formal mathematical definition of how the limit of a function at a point is formed. It is sometimes called the precise or formal definition of the limit. Here is what it looks like.

\(\epsilon\)-\(\delta\) Limit Definition

Let f be a function defined on an open interval containing c (except possibly at c) and let L be a real number. The statement \[ \lim_{x \rightarrow c}{~f(x)}=L \] means that for each \( \epsilon \gt 0 \) there exists a \( \delta \gt 0 \) such that if \( 0 \lt \left|x-c\right| \lt \delta\), then \(\left|f(x)-L\right| \lt \epsilon .\)

This is called the epsilon-delta definition of the limit because of the use of \(\epsilon\) (epsilon) and \(\delta\) (delta) in the text above. This is standard notation that most mathematicians use, so you need to use it as well. See the Use Of The Greek Alphabet in Mathematics section on the notation page for more information.

Something that may not be obvious from the definition is that \(\delta\) depends on \(\epsilon\), i.e. \(\delta = M(\epsilon)\) where \(M\) is some function. The third video below emphasizes this and shows details on how this works.

This definition is not easy to get your head around and it takes some thinking, working practice problems and time. I highly recommend you watch this video. It will help you get a feel for the concept of this definition.

Krista King Math - Limit Definition [6min-41secs]

video by Krista King Math

Okay, after watching the above video, you should have at least an idea of what the epsilon-delta definition of the limit means. Now let's watch a video that discusses this in more depth.

PatrickJMT - Precise Definition of a Limit - Understanding the Definition [11min-5secs]

video by PatrickJMT

This next video is a great video showing an example. Normally, we would make a practice problem from this video. However, this video has some excellent explanation that we don't want you to miss.

Dr Chris Tisdell - Limit Definition [6min-22secs]

In this video, he talks about the \(\epsilon\)-\(M\) definition and describes \(M\) as a function of \(\epsilon\). His \(M\) is what we call \(\delta\) and our experience is that most books and instructors use \(\delta\). However, his use of \(M(\epsilon)\) is good since it emphasizes that \(\delta\) is a function of (and depends on) \(\epsilon\).

video by Dr Chris Tisdell

Okay, time for a few practice problems. After that, you will be ready for the next topic, finite limits.

next: finite limits →

Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. Here is a list converting the old numbering system to the new.

Practice Problems Conversion

[A01-447] - [A02-2014] - [A03-2017] - [B01-448] - [B02-449] - [B03-450]

Please update your notes to this new numbering system. The display of this conversion information is temporary.

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Basic Problems

Use the \(\epsilon\)-\(\delta\) definition of the limit to prove \(\displaystyle{\lim_{x\to-3}{(7x-9)}=-30}\).

Problem Statement

Use the \(\epsilon\)-\(\delta\) definition of the limit to prove \(\displaystyle{\lim_{x\to-3}{(7x-9)}=-30}\).

Solution

447 solution video

video by Krista King Math

close solution

Use the \(\epsilon\)-\(\delta\) definition of the limit to prove \(\displaystyle{\lim_{x\to 4}{(2x+3)} = 11}\)

Problem Statement

Use the \(\epsilon\)-\(\delta\) definition of the limit to prove \(\displaystyle{\lim_{x\to 4}{(2x+3)} = 11}\)

Solution

2014 solution video

video by PatrickJMT

close solution

Use the \(\epsilon\)-\(\delta\) definition of the limit to prove \(\displaystyle{\lim_{x\to 2}{(x^2-4x+5)}=1}\)

Problem Statement

Use the \(\epsilon\)-\(\delta\) definition of the limit to prove \(\displaystyle{\lim_{x\to 2}{(x^2-4x+5)}=1}\)

Solution

2017 solution video

video by PatrickJMT

close solution

Intermediate Problems

Given the limit \(\displaystyle{\lim_{x\to1}{(2x+3)}=5}\), find the largest value of \(\delta\) such that \(\epsilon < 0.01 \).

Problem Statement

Given the limit \(\displaystyle{\lim_{x\to1}{(2x+3)}=5}\), find the largest value of \(\delta\) such that \(\epsilon < 0.01 \).

Solution

448 solution video

video by PatrickJMT

close solution

Given the limit \(\displaystyle{\lim_{x\to-1}{(3x+1)}=-2}\), find the largest value of \(\delta\) such that \(\epsilon=0.01\).

Problem Statement

Given the limit \(\displaystyle{\lim_{x\to-1}{(3x+1)}=-2}\), find the largest value of \(\delta\) such that \(\epsilon=0.01\).

Solution

449 solution video

video by PatrickJMT

close solution

Find \(\displaystyle{\lim_{x\to1}{\frac{3x(x-1)}{x-1}}}\) and prove your answer using the \(\epsilon\)-\(\delta\) definition.

Problem Statement

Find \(\displaystyle{\lim_{x\to1}{\frac{3x(x-1)}{x-1}}}\) and prove your answer using the \(\epsilon\)-\(\delta\) definition.

Solution

450 solution video

video by Khan Academy

close solution
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