- https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_limit
- This wikipedia article provides the history and a brief explanation of the epsilon-delta definition of the limit of a function.
- https://www.khanacademy.org/math/ap-calculus-ab/limits-basics-ab/formal-definition-of-limits-ab/v/epsilon-delta-definition-of-limits
- In this 3rd video of a series on the limit of a function, Khan goes into some depth to derive the epsilon-delta definition of the limit of a function.
- Excercises:
- This document from UC Berkely uses the epsilon-delta definiton of the limit of a function to prove the limits of four different functions.
4A: The Limit of a Function (Part 1) (Page 263 – 282)
By Allen Charest
In your calculus class, you were introduced to the idea of what a limit is. Normally, it is set up as: lim x→x0 f(x). But by looking at it this way, we are not able to fully understand the idea of the limit. But this term can be broadened which can help us understand how real analysis works.
- Limits (ε-δ Definition)
One of the terms we can broaden is the x0. In this class, we are able to call that point the accumulation point. And by using this different term, we can then create a new definition of limits which will incorporate other useful terms. Some of these include epsilon (ε), an arbitrarily small and positive quantity and delta (δ) which implies a difference or change.
Definition 5.1: (Limit) Let f : E → R be a function with domain E and suppose that x0 is a point of accumulation of E. Then we write limx→x0 f(x) = L if for every ε > 0 there is a δ > 0 so that: |f(x) − L| < ε
whenever x is a point of E differing from x0 and satisfying |x − x0| < δ
The condition on x can be written as
0 < |x − x0| < δ
or as
x ∈ (x0 − δ, x0 + δ)
Here is a simple example to show how this new definition works:
Let us do this for the linear function f(x) = 10x − 11. We expect that
Lim x→5 (10x − 11) = 10(5) − 11 = 39.
Let us prove this. We need a condition ensuring that the expression
|(10x − 11) − 39| < ε. Some arithmetic converts this to
|(10x − 11) − 39| = |10x − 50| = |10| |x − 5| .
Now it is clear that, if we insist that |x − 5| < ε/10, we will have
|(10x − 11) − 39| < ε.
With this information, we now have everything to write the proof:
Let ε > 0. Let δ = ε/10. Then for all x with |x − 5| < δ we have
|(10x − 11) − 39| = |10| |x − 5| < 10δ = ε.
By definition, limx→5(10x − 11) = 39 as required.
Here are some additional resources to understand how these limits work:
Defining the Limit of a Sequence
https://www.youtube.com/watch?v=Nj-VTJecLn8&list=PLL0ATV5XYF8B0C2UU7J_JbeuGvugN_P56&index=10
- Limits Sequential Definition:
One of the other things we are able to do with this new thinking of limits is create new definitions that are more based upon the sequence itself. It provides us with a new perspective of seeing how we find a limit. This new thinking can be shown this way:
Definition 5.4: (Limit) Let f : E → R be a function with domain E and suppose that x0 is a point of accumulation of E. Then we write
limx→x0 f(x) = L if for every sequence {en} of points of E with en 6= x0 and en → x0 as n → ∞,
limn→∞ f(en) = L
- Limits (Mapping Definition):
Even though each of the categories seem different from one another, they are actually equivalent to one another. They all show different and more in-depth perspectives on how to prove the limit of a sequence. This one a mapping property that can be shown by open sets.
Definition 5.6: (Limit) Let f : E → R be a function with domain E and suppose that x0 is a point of accumulation of E. Then we write
limx→x0 f(x) = L if for every open set V containing the point L there is an open set U containing the point x0 and every point x 6= x0 of U that is in the domain of f is mapped into a point in V ;
that is, f : E ∩ U \ {x0} → V.
- One-Sided Limits:
Now that we have established many different ways to understand one idea, we can take these ways and apply them to a new idea. This time, we can apply them to the idea of one-sided limits. In case you need a refresher, here is a short video with some examples of them: https://www.youtube.com/watch?v=Rp4e_RySr0g. It is basically having to look at a limit from one side of an accumulation point or the other. And this limit can be seen in the same way that we have shown regular limits. Here are some definitions of one-sided limits to give you the idea of how they are similar:
Definition 5.7: (Right-Hand Limit) Let f : E → R be a function with domain E and suppose that x0 is a point of accumulation of E ∩ (x0, ∞). Then we write
lim x→x0+ f(x) = L if for every ε > 0 there is a δ > 0 so that
|f(x) − L| < ε whenever x0 < x < x0 + δ and x ∈ E
Here is another way we can think of this same right-hand limit idea:
Definition 5.8: (Right-Hand Limit) Let f : E → R be a function with domain E and suppose that x0 is a point of accumulation of E ∩ (x0, ∞). Then we write
lim x→x0+ f(x) = L if for every decreasing sequence {en} of points of E with en > x0 and en → x0 as n → ∞,
limn→∞ f(en) = L.
- Infinite Limits:
The next thing we will look at from a different perspective is a limit that continues up to infinity. With the terms and definitions we already know, we can create this definition:
Definition 5.9: (Infinite Limit) Let f : E → R be a function with domain E and suppose that x0 is a point of accumulation of E ∩ (x0, ∞). Then we write:
lim x→x0+ f(x) = ∞ if for every M > 0 there is a δ > 0 so that:
f(x) ≥ M whenever x0 < x < x0 + δ and x ∈ E.
There are also a few different ways to restructure this definition. For more insight on these different ways, look at this source:
https://www.youtube.com/watch?v=pxE04Gb8cXY
- Properties of Limits:
The last major topic I will be talking about is the property of a limit. Before in calculus class, it was easy enough to see how to add, subtract, multiply and divide functions. But now, we are able to come up with the basic theories of functions and their limits. The first one we will look at is the uniqueness of a limit. When we write that a function approaches a certain limit, we want to make sure that it is not true for values different than that limit. Here is how the definition would look like:
Theorem 5.10 (Uniqueness of Limits) Suppose that limx→x0 f(x) = L.
Then the number L is unique: No other number has this same property.
Suppose we had the same function approaching different limit values. For this to happen, they must have the same accumulation point. But this would end up being impossible unless you have the same limit value. For more about this, look at:
https://www.youtube.com/watch?v=A4LhIhI6tg4
The next thing we will look at is how a function is bounded. For a reminder of how the boundedness of a limit works, look at:
https://www.youtube.com/watch?v=wasEnmn-mFM
If the limit of a function exists in the first place, there must be restrictions so that the limit does not go any higher than the value of that limit. Even for functions that go out to infinity, the function may seem unbounded but they are locally bounded at each point other than where they approach. Here is what the definition looks like:
Theorem 5.11 (Boundedness of Limits) Suppose that the limit limx→x0 f(x) = L exists.
Then there is an interval (x0 − c, x0 + c) and a number M such that |f(x)| ≤ M for every value of x in that interval that is in the domain of f.
The last thing we will look at is how a function can be bounded even when it is going away from zero. If a function ends up having a nonzero limit, then the function will be away from zero as it is approaching whatever nonzero limit it is. Here is how the structure of this definition looks:
Theorem 5.12 (Boundedness Away from Zero)
If the limit limx→x0 f(x) exists and is not zero, then there is an interval (x0 − c, x0 + c) and a positive number m such that |f(x)| ≥ m > 0 for every value of x 6= x0 in that interval and that belongs to the domain of f.
If you look back on everything we have gone over, you can see that a lot of it comes from ideas we have already done, like limits, functions, and how they are bounded. The only thing we are doing now is expanding our ideas on each of these topics and seeing how they can be learned from different perspectives.
4A: The Limit of a Function (Part 2) (Page 282 – 305)
By Tori Fortune
I. Algebra of Limits
Thinking back to Calculus class, you may remember some properties of limits such as “the limit of a sum is equal to the sum of the limits.” Well, these algebraic properties are still applicable while exploring the limits of functions in Analysis.
· Definitions
1. Multiples of Limits: Let lim𝑥→𝑎 𝑓(𝑥) and C ∈ℝ exist. Then, lim𝑥→𝑎 C𝑓(𝑥) = C (lim𝑥→𝑎 𝑓(𝑥)).
Þ i.e. The limit of the product of a constant and a function is equal to the product of that constant and the limit of that function. i.e. The constant can be “taken out.”
2. Sums and Differences of Limits: Let lim𝑥→𝑎 𝑓(𝑥) and lim𝑥→𝑎 𝑔(𝑥) exist and let a be an accumulation point of dom(𝑓)∩ dom(𝑔). Then, lim𝑥→𝑎 (𝑓(𝑥)+𝑔(𝑥)) = lim𝑥→𝑎 𝑓(𝑥) + lim𝑥→𝑎 𝑔(𝑥) and , lim𝑥→𝑎 (𝑓(𝑥)−𝑔(𝑥)) = lim𝑥→𝑎 𝑓(𝑥) – lim𝑥→𝑎 𝑔(𝑥).
Þ i.e. The sum of the limits of two functions is equal to the limit of the sum of the two functions.
3. Products of Limits: Let lim𝑥→𝑎 𝑓(𝑥) and lim𝑥→𝑎 𝑔(𝑥) exist and let a be an accumulation point of dom(𝑓)∩ dom(𝑔). Then, lim𝑥→𝑎 𝑓(𝑥)𝑔(𝑥) = (lim𝑥→𝑎 𝑓(𝑥))(lim𝑥→𝑎 𝑔(𝑥)).
Þ i.e. The product of the limits of two functions is equal to the limit of the product of those two functions.
4. Quotients of Limits: Let lim𝑥→𝑎 𝑓(𝑥) and lim𝑥→𝑎 𝑔(𝑥) exist, where lim𝑥→𝑎 𝑔(𝑥)≠0. Let a be an accumulation point of dom(𝑓)∩ dom(𝑔). Then, lim𝑥→𝑎 𝑓(𝑥)𝑔(𝑥)= lim𝑥→𝑎 𝑓(𝑥) lim𝑥→𝑎 𝑔(𝑥).
Þ i.e. The quotient of the limits of two functions is equal to the limit of the quotient of those two functions, excluding the case where the limit of the second function is equal to 0.
· Helpful Source(s) 1. https://www.youtube.com/watch?v=TK_K_rM0N00
This seventeen-minute video can help you better understand the above mentioned limit laws. There is a proof for each of the definitions along with proofs for other properties that are not mentioned on this page. 2. https://www.youtube.com/watch?v=jYQvaVOafxI
This video shows a proof of the product of limits definition. For those who prefer a more visual explanation, the video provides a graphic view of what happens to the limit when you take the product of two functions.
· Example Problem
1. Let C = 4, 𝑙𝑖𝑚𝑥→𝑎 𝑓(𝑥)= 0, 𝑙𝑖𝑚𝑥→𝑎 𝑔(𝑥)=12, where a is an accumulation point of dom(𝑓)∩ dom(𝑔). Find the following:
a. 𝑙𝑖𝑚𝑥→𝑎 𝐶𝑔(𝑥)
b. 𝑙𝑖𝑚𝑥→𝑎 (𝑓(𝑥)−𝑔(𝑥))
c. 𝑙𝑖𝑚𝑥→𝑎 𝑓(𝑥)𝑔(𝑥)
d. 𝑙𝑖𝑚𝑥→𝑎 𝐶(𝑔(𝑥)𝑓(𝑥))
2. Use the epsilon-delta definition of a limit to prove the “Multiples of Limits” theorem.
II. Order Properties
Not only can we understand the limits of combinations of functions, but we can also understand the relationship between limits of functions by drawing from the functions themselves. The order properties of limits help us to do that.
· Definitions
1. Order of Limits: Let lim𝑥→𝑎 𝑓(𝑥) and lim𝑥→𝑎 𝑔(𝑥) exist and let a be an accumulation point of dom(𝑓)∩ dom(𝑔). If 𝑓(𝑥)≤𝑔(𝑥) for all x ∈𝑑𝑜𝑚(𝑓)∩ 𝑑𝑜𝑚(𝑔), then, lim𝑥→𝑎 𝑓(𝑥)≤lim𝑥→𝑎 𝑔(𝑥).
Þ i.e. If the first function is less than or equal to the second function, then it follows that the limit of the first function is less than or equal to the limit of the second function.
2. Corollary for Order of Limits: Let lim𝑥→𝑎 𝑓(𝑥) exist and let 𝛼≤𝑓(𝑥)≤𝛽 for all x in 𝑑𝑜𝑚(𝑓). Then, 𝛼≤lim𝑥→𝑎 𝑓(𝑥)≤𝛽.
Þ i.e. If a function is greater than or equal to a number, alpha, and less than or equal to a number, beta, then the limit of the function is greater than or equal to alpha and less than or equal to beta.
3. Squeeze Theorem: Let f, g, h: E →ℝ exist and let a be an accumulation point of dom(𝑓)∩ dom(𝑔)∩ dom(h)= E. Suppose that the limits lim𝑥→𝑎 𝑓(𝑥)= L and lim𝑥→𝑎 𝑔(𝑥)= L exist
and that f(x) ≤ h(x) ≤ g(x) for all x ∈𝐸 except perhaps at x=a. Then lim𝑥→𝑎 ℎ(𝑥)= L.
Þ i.e. If the limit of two different functions are equal, then the limit of a function which is “in between” the two functions has the same limit as well.
4. Limits of Absolute Values: Let the limit lim𝑥→𝑎 𝑓(𝑥)= L exist. Then, lim𝑥→𝑎 |𝑓(𝑥)| = |L|.
Þ i.e. The limit of the absolute value of a function is equal to the absolute value of the limit.
5. Maximum and Minimum of Limits: Let the limits lim𝑥→𝑎 𝑓(𝑥)= L and lim𝑥→𝑎 𝑔(𝑥)= M exist and let a be an accumulation point of dom(𝑓)∩ dom(𝑔). Then, lim𝑥→𝑎 max{𝑓(𝑥), 𝑔(𝑥)} = max{L, M} and lim𝑥→𝑎 min{𝑓(𝑥), 𝑔(𝑥)} = min{L, M}.
Þ i.e. The limit of the maximum of two functions is the maximum of the limits of those two functions and the limit of the minimum of two functions is the minimum of the limits of those two functions.
· Helpful Source(s) 1. https://www.youtube.com/watch?v=WvxKwRcHGHg
This Khan Academy video uses an analogy to offer an alternative explanation of the squeeze theorem. It is helpful in understanding the definition of this theorem by breaking it down into simpler terms and presenting a proof.
· Example Problem(s)
1. Let 𝛼≤ f(x) ≤ h(x) ≤𝛽 ≤ g(x) be true and let a be an accumulation point of dom(𝑓)∩ dom(𝑔)∩ dom(h). Which of the following are true and which are false? If false, explain why.
a. lim𝑥→𝑎 𝑓(𝑥)≤lim𝑥→𝑎 𝑔(𝑥)
b. lim𝑥→𝑎 ℎ(𝑥)≤lim𝑥→𝑎 𝑓(𝑥)
c. 𝛼≤lim𝑥→𝑎 𝑔(𝑥)≤𝛽
d. 𝛼≤lim𝑥→𝑎 ℎ(𝑥)≤𝛽
e. If lim𝑥→𝑎 𝑓(𝑥)= L-1 and lim𝑥→𝑎 𝑔(𝑥)= L+1 then lim𝑥→𝑎 ℎ(𝑥)=𝐿
f. If lim𝑥→𝑎 𝑓(𝑥)=−1 then lim𝑥→𝑎 |𝑓(𝑥)|=1
g. If lim𝑥→𝑎 𝑓(𝑥)=0 and lim𝑥→𝑎 ℎ(𝑥)=1, then lim𝑥→𝑎 max{𝑓(𝑥), ℎ(𝑥)}= 0
2. Give a proof of the “Limit of Absolute Values” theorem by converting it into a statement about sequences.