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You CAN Ace Calculus

17calculus > limits > epsilon-delta definition

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Math Word Problems Demystified

Epsilon-Delta Definition of the Limit

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 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

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

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.
After that, you will be ready for the next topic, finite limits.

next: finite limits →

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Practice Problems

Level A - Basic

Practice A01

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


Practice A02

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


Practice A03

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


Level B - Intermediate

Practice B01

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


Practice B02

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


Practice B03

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


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