Squeeze Theorem
Illustration of the example
The squeeze theorem is a theorem used in calculus to evaluate a limit of a function.
The theorem is particularly useful to evaluate limits where other techniques might be unnecessarily complicated. For example, is somewhat tricky as it is the product of a term that goes to 0 and one that does not converge. However, since and and both tend to as it must be the case that
To the right, the graph of the function under study is in red.
Contents
Statement of the Theorem and its Intuition
Intuition: Another name for the squeeze theorem is the sandwich theorem. Notice how the theorem makes a "sandwich" of function between functions and on a subset of . and have been "cleverly chosen" (if possible) to have the same limit as . Since and enclose the three functions must all have the same limit.
Assume that functions defined in satisfy
Then, if then
Proof of the Theorem
This theorem is equivalent to the next theorem: If three sequences fulfill with and have the same limit , then has the same limit , due to if and only if is .
We are going to prove the last theorem: Given an which implies
Identifying and Solving Limits that Require Squeeze Theorem
1. Upper and lower bounds explicitly given
When the upper bound and the lower bound are explicitly given in the problem, you just need to transform the squeezed one into the one you're trying to find the value of.
Given an infinite sequence that satisfies for all positive integers evaluate
Note that sending to infinity without transforming the inequality would result in a catastrophe.We will transform the middle term of the inequality into the desired expression:
Now we can send to infinity to obtain
Hence we conclude that
This method is often used in sequences, and this type of problems is well-known in high school.
2. Absolute value (usually the function value heads to 0)
There are many things that you could try when you see an absolute value in a problem, and the squeeze theorem is one of them.
Usually, the start is surrounding the desired expression by utilizing the properties of an absolute value.
There is a function Prove that if then
When dealing with this problem, you need to understand the following property of an absolute value:
This can be easily checked via letting and for respective situations.
Then, we utilize this property to surround the desired expression:
Note that
This is enough to conclude that
3. Trigonometric function (I)
Sometimes it dazzles the student when a weird problem with trigonometric functions in limits or sums arises like the following:
Evaluate the limit
Note that
So, we can write
The leftmost side is automatically while
This is sufficient to conclude that
4. Sum with a limit that tends to infinity
When you see a sum within a limit that tends to infinity, sometimes it is wise to use the squeeze theorem.
Evaluate the limit
We observe that so thatWe also note that for it is always true that
Then, we proceed to find the upper bound and the lower bound:
Now we can have
The leftmost expression converges to
The rightmost expression also converges to
This is sufficient to conclude that
5. Trigonometric function (II)
It often is the case that constant bounds just aren't enough. We then would need to find ourselves some brand-new bounds using trigonometric identities and geometry.
An infinite sequence satisfies
for all positive integers Prove that
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Think as here. We can have this because for all
Since the area of sector is in between the areas of and we can have
Manipulating this inequality yields
Noting that
we can say that
Now we need to observe this:
From the original inequality,
where the leftmost and rightmost expressions both approach as tends to infinity. Therefore,
6. Functional equations
In this type of problems, you will need some clever manipulation and/or substitution to figure out what the upper bound and the lower bound are.
Given a function that satisfies
for all reals and prove that is a constant function.
Interchange and to obtain
so we can see that
Set and divide both sides by :
Set and divide both sides of by :
The two inequalities are exactly the same, so we proceed to the next step:
So,
Therefore, for all reals which proves that is a constant function.
7. Miscellaneous
Sometimes there just isn't any pattern.
An infinite sequence with positive terms satisfies
for all positive integers Prove that approaches as tends to infinity.
It is crucial to note that the value of is always within the rangeSince
we can see that
and since the right side approaches as tends to infinity, it is obvious that
Determining Appropriate Lower and Upper Bounds
Examples
Evaluate
Since
Note that for which implies
By the squeeze theorem, we get
Evaluate
Let then
Note that which implies
Evaluate
Let then
Note that
which implies
Use the squeeze theorem to prove that .
When is close to 0, we have
Note that which implies
Using the squeeze theorem, evaluate
Note that
As and get closer and closer to as
Thus,
Evaluate
We know that which implies
Note that so
Find
We start with the inequality that is pretty much the definition of the radian measure of the angles. Dividing by yields But and we can apply the squeeze theorem. We assumed to be positive, but is even, so we are done. Actually, we could simply take instead of using the squeeze theorem to finish the proof.
For , define as above.
Evaluate