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EpsilonDelta Definition of the Limit 

The epsilondelta 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 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 \( 0 \lt \leftxc\right \lt \delta\), then \(\leftf(x)L\right \lt \epsilon .\) 
This is called the epsilondelta 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  
Okay, after watching the above video, you should have at least an idea of what the epsilondelta 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  
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  
Okay, time for a few practice problems.  next: finite limits → 
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Practice Problems 

Level A  Basic 
Practice A01  

Use the \(\epsilon\)\(\delta\) definition of the limit to prove \(\displaystyle{\lim_{x\to3}{(7x9)}=30}\).  
solution 
Practice A02  

Use the \(\epsilon\)\(\delta\) definition of the limit to prove \(\displaystyle{\lim_{x\to 4}{(2x+3)} = 11}\)  
solution 
Practice A03  

Use the \(\epsilon\)\(\delta\) definition of the limit to prove \(\displaystyle{\lim_{x\to 2}{(x^24x+5)}=1}\)  
solution 
Level B  Intermediate 
Practice B01  

Given the limit \(\displaystyle{\lim_{x\to1}{(2x+3)}=5}\), find the largest value of \(\delta\) such that \(\epsilon < 0.01 \).  
solution 
Practice B02  

Given the limit \(\displaystyle{\lim_{x\to1}{(3x+1)}=2}\), find the largest value of \(\delta\) such that \(\epsilon=0.01\).  
solution 
Practice B03  

Find \(\displaystyle{\lim_{x\to1}{\frac{3x(x1)}{x1}}}\) and prove your answer using the \(\epsilon\)\(\delta\) definition.  
solution 