You CAN Ace Calculus
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Single Variable Calculus |
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Multi-Variable Calculus |
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Acceleration Vector |
Arc Length (Vector Functions) |
Arc Length Function |
Arc Length Parameter |
Conservative Vector Fields |
Cross Product |
Curl |
Curvature |
Cylindrical Coordinates |
Lagrange Multipliers |
Line Integrals |
Partial Derivatives |
Partial Integrals |
Path Integrals |
Potential Functions |
Principal Unit Normal Vector |
Differential Equations |
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Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.
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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 |
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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.
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.
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.
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 → |
Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems |
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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. |
Basic Problems |
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Use the \(\epsilon\)-\(\delta\) definition of the limit to prove \(\displaystyle{\lim_{x\to-3}{(7x-9)}=-30}\).
Problem Statement |
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Use the \(\epsilon\)-\(\delta\) definition of the limit to prove \(\displaystyle{\lim_{x\to-3}{(7x-9)}=-30}\).
Solution |
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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 |
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Use the \(\epsilon\)-\(\delta\) definition of the limit to prove \(\displaystyle{\lim_{x\to 4}{(2x+3)} = 11}\)
Solution |
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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 |
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Use the \(\epsilon\)-\(\delta\) definition of the limit to prove \(\displaystyle{\lim_{x\to 2}{(x^2-4x+5)}=1}\)
Solution |
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video by PatrickJMT
close solution |
Intermediate Problems |
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Given the limit \(\displaystyle{\lim_{x\to1}{(2x+3)}=5}\), find the largest value of \(\delta\) such that \(\epsilon < 0.01 \).
Problem Statement |
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Given the limit \(\displaystyle{\lim_{x\to1}{(2x+3)}=5}\), find the largest value of \(\delta\) such that \(\epsilon < 0.01 \).
Solution |
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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 |
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Given the limit \(\displaystyle{\lim_{x\to-1}{(3x+1)}=-2}\), find the largest value of \(\delta\) such that \(\epsilon=0.01\).
Solution |
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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 |
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Find \(\displaystyle{\lim_{x\to1}{\frac{3x(x-1)}{x-1}}}\) and prove your answer using the \(\epsilon\)-\(\delta\) definition.
Solution |
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video by Khan Academy
close solution |