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17Calculus - Epsilon-Delta (Precise or Formal) Definition of the Limit

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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.

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. 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. This video emphasizes this and shows details on how this works.

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, one more video. Here is a great explanation of the definition with some hand-drawn graphs and one example that explains in excruciating detail. But that detail is very important and he explains it very well.

Jeremy Klassen - An Introduction to Limits

video by Jeremy Klassen

Limit Key

One key that you need to remember about limits is, when you use the limit notation \[ \lim_{x \rightarrow c}{ ~f(x) } \] this means that \(x\) APPROACHES \(c\) but IS NEVER EQUAL TO \(c\). That is, \(x\) can get as close as it wants to \(c\) but it will never actually equal \(c\). That seems simple enough but it is extremely important to remember.

However, there are many times when you can determine the limit by substituting x for c to calculate f(c). However, those are special cases that require special conditions and is not true in every case.

Here is an explanation of how this concept is written mathematically. Look more carefully at the definition of the limit at the top of the page. Notice that it requires \( \delta \gt 0 \). This means that \( |x-c| > 0 \) and, therefore, x can never equal c.

I know this seems like a minor point, but it isn't. If you remember this, you will have a good start on your way to understanding limits.

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

Calculus, Better Explained: A Guide To Developing Lasting Intuition

Practice

Unless otherwise instructed, use the \(\epsilon\)-\(\delta\) definition of the limit to prove the limit exists. If the limit is not given, find it first.

Basic

\(\displaystyle{ \lim_{x \to 5}{ (3x-1) } = 14 }\)

Problem Statement

Use the \(\epsilon\)-\(\delta\) definition of the limit to prove \(\displaystyle{ \lim_{x \to 5}{ (3x-1) } = 14 }\)

Solution

3145 video solution

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\(\displaystyle{ \lim_{x \to 2}{ (3x+5) } = 11 }\)

Problem Statement

Use the \(\epsilon\)-\(\delta\) definition of the limit to prove \(\displaystyle{ \lim_{x \to 2}{ (3x+5) } = 11 }\)

Solution

3146 video solution

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\(\displaystyle{ \lim_{x \to 8}{ (3x+5) } = 29 }\)

Problem Statement

Use the \(\epsilon\)-\(\delta\) definition of the limit to prove \(\displaystyle{ \lim_{x \to 8}{ (3x+5) } = 29 }\)

Solution

3147 video solution

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\(\displaystyle{ \lim_{x \to -4}{ (2x-3) } = -11 }\)

Problem Statement

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

Solution

3148 video solution

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\(\displaystyle{ \lim_{x \to 5}{ (4x-7) } = 13 }\)

Problem Statement

Use the \(\epsilon\)-\(\delta\) definition of the limit to prove \(\displaystyle{ \lim_{x \to 5}{ (4x-7) } = 13 }\)

Solution

turksvids - 3150 video solution

video by turksvids

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\(\displaystyle{\lim_{x\to-3}{(7x-9)}=-30}\)

Problem Statement

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

Solution

Krista King Math - 447 video solution

video by Krista King Math

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\(\displaystyle{\lim_{x\to 4}{(2x+3)} = 11}\)

Problem Statement

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

Solution

PatrickJMT - 2014 video solution

video by PatrickJMT

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\(\displaystyle{ \lim_{x \to -2}{ (x/4+6) } = 11/2 }\)

Problem Statement

Use the \(\epsilon\)-\(\delta\) definition of the limit to prove \(\displaystyle{ \lim_{x \to -2}{ (x/4+6) } = 11/2 }\)

Solution

turksvids - 3149 video solution

video by turksvids

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Intermediate

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

PatrickJMT - 448 video solution

video by PatrickJMT

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

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

PatrickJMT - 449 video solution

video by PatrickJMT

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\(\displaystyle{\lim_{x\to1}{\frac{3x(x-1)}{x-1}}}\)

Problem Statement

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

Solution

Khan Academy - 450 video solution

video by Khan Academy

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Advanced

\(\displaystyle{ \lim_{x \to 2}{ (x^2+3x-9) } = 1 }\)

Problem Statement

Use the \(\epsilon\)-\(\delta\) definition of the limit to prove \(\displaystyle{ \lim_{x \to 2}{ (x^2+3x-9) } = 1 }\)

Solution

turksvids - 3151 video solution

video by turksvids

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\(\displaystyle{ \lim_{x \to 3}{ (2x^2-3x+1) } = 10 }\)

Problem Statement

Use the \(\epsilon\)-\(\delta\) definition of the limit to prove \(\displaystyle{ \lim_{x \to 3}{ (2x^2-3x+1) } = 10 }\)

Solution

turksvids - 3152 video solution

video by turksvids

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\(\displaystyle{\lim_{x\to 2}{(x^2-4x+5)}=1}\)

Problem Statement

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

Solution

PatrickJMT - 2017 video solution

video by PatrickJMT

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Definition

Limit Key

next: finite limits →

next: finite limits →

Practice Instructions

Unless otherwise instructed, use the \(\epsilon\)-\(\delta\) definition of the limit to prove the limit exists. If the limit is not given, find it first.

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