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The Squeeze Theorem

The Squeeze Theorem

Calculus presents us with various useful computational tricks, especially in the field of limits. When faced with oscillating functions or functions with undefined points, taking the limit can become a difficult task. Luckily, The Squeeze, or Sandwich, Theorem is just the trick for dealing with tricky functions such as these.

The Squeeze Theorem Definition

The Squeeze Theorem is a limit evaluation method where we "squeeze" an indeterminate limit between two simpler ones. The "squeezed" or "bounded" function approaches the same limit as the other two functions surrounding it.

More precisely, the Squeeze Theorem states that for functions f, g, and h such that , if

for a constant L, then

.The Squeeze Theorem visual intuition behind Squeeze Theorem StudySmarterAs f(x) is "squeezed" between g(x) and h(x), the Squeeze Theorem can be applied to evaluate the limit of f(x) at x = 0 - StudySmarter Originals

The Squeeze Theorem Proof

Informal Proof

Simply put, is "squeezed" betweenand. Asand are equal at the point A such that , then , as there is no room between the other two functions for f to take on any other value.

Formal Proof

We will assume that

  • everywhere in the domain of the functions

Upon these assumptions, we want to prove that

.

See the image below for a visual explanation of the variables!

Let an arbitrary epsilon such that be known. To prove the Squeeze Theorem, we must find a delta such that whenever where L is the evaluation of the limit as x approaches the point A.

Now by definition, so there must exist some such that

for all

Using absolute value laws

(1) for all.

Similarly, by definition, so there must exist some such that

for all .

Using absolute value laws

(2) for all .

Squeeze Theorem proof geometric visual StudySmarterVisual explanation of the geometric derivation of (1) and (2) - StudySmarter Original

Since for all x on some open interval containing A, there must exist some such that

(3) for all

Where, then by (1), (2), and (3)

for all

Thus,

for all

Using absolute value laws

for all

Then, by definition

When to use the Squeeze Theorem Formula

The Squeeze Theorem should be used as a last resort. When solving limits, one should always try to solve through algebraic or simple manipulation first. If algebra fails, the Squeeze Theorem may be a viable option for limit solving.

Indeed, to calculate , we must first find two functions and that bound and such that

The Squeeze Theorem cannot be applied if the limits of the bounding functions are not equal.

Examples of Evaluating Limits using the Squeeze Theorem

Let's start with a simple example!

Use the Squeeze Theorem to evaluate

When we plug in x = 0, we are met with an undefined form . This is a perfect candidate for the Squeeze Theorem!

This is an example of a general scenario: the Squeeze Theorem can be applied to find the limit of trigonometric functions damped by polynomial terms.

The general strategy for solving these kinds of Squeeze Theorem problems is to- start with the trigonometric function, in this case, - build up to the function in the problem; here it isLet's see how this is done!
  • Step 1: Make a double-sided inequality to bound the trigonometric function based on the nature of the cosine function.
    • We know that the cosine function oscillates on the closed interval [-1, 1], i.e.
    • In graphing, we find that f also oscillates on the closed interval [-1, 1], i.e.

It is essential to know that the range of cos (anything) and sin (anything) will always be [-1, 1] (as long as it is not translated up/down or vertically stretched/compressed)!

Squeeze Theorem cosine graph StudySmarterExample 1 graph - StudySmarter Originals

  • Step 2: Modify the inequality as needed to bound the function of the problem:
    • Our function is so we multiply our double-sided inequality byand get

  • Step 3: Verify that the bounding functions have the same limit.
    • Now that our function is bounded, we must verify thatin order to apply the Squeeze Theorem:

  • Step 4: Apply the Squeeze Theorem
    • Since , then by the Squeeze Theorem

Now, let's try something a bit more complex.

Find

When we plug in, we are left with the indeterminate form . Again, since a trigonometric function appears, this is a perfect candidate for the Squeeze Theorem!Following the same strategy as before, start with the trigonometric function , and build up to
Let's take a look!
  • Step 1: Make a double-sided inequality to bound the trigonometric function based on the nature of the sine function.
    • We know that sine behaves like cosine in that it oscillates on the closed interval [-1, 1].
    • Looking at the image below, when we graph , we find that

.

Squeeze Theorem sin(5x) graph StudySmarter
Example 2 graph - StudySmarter Original
  • Step 2: Modify the inequality as needed to bound the function of the problem
    • Our function is and our double-sided inequality is
    • Add to get
    • Multiply by to get
  • Step 3: Now that our function is bounded, we must verify that in order to apply the Squeeze Theorem.
    • Again, when we try to plug in , we are met with an indeterminate form. However, this time we can use algebraic manipulation to solve.
    • Multiply both limits by to get and
    • Now, when we plug in , we get and respectively leaving

  • Step 4: Apply the Squeeze Theorem
    • Since , then by the Squeeze Theorem,

The Squeeze Theorem - Key takeaways

  • The Squeeze Theorem is a last resort method for solving limits that cannot be solved through algebraic manipulation.
  • The Squeeze Theorem states that for functions f, g, and h such that , if

for a constant L, then

  • If , the Squeeze Theorem cannot be applied
  • The general strategy for solving problems containing trigonometric functions is to start with the trig function, then build up to the function in the question!
  • A step-by-step procedure for the Squeeze Theorem is:
    • Step 1: Make a double-sided inequality based on the nature of f(x)
    • Step 2: Modify the inequality as needed
    • Step 3: Solve the limits on both sides of the inequality, ensuring that they are equal
    • Step 4: Apply the Squeeze Theorem - the limit of f(x) is equal to the bounding limits

Frequently Asked Questions about The Squeeze Theorem

The Squeeze Theorem is a method for solving limits that cannot be solved through algebra or other simple manipulations.

To solve with the Squeeze Theorem:

  • Make a double-sided inequality based on the nature of f(x)
  • Modify the inequality as needed
  • Solve the limits on both side of the inequality (they must be equal to apply the Squeeze Theorem)
  • The limit of f(x) is equal to the limits of the outside limits

To solve with the Squeeze Theorem:

  • Make a double-sided inequality based on the nature of f(x)
  • Modify the inequality as needed
  • Solve the limits on both side of the inequality (they must be equal to apply the Squeeze Theorem)
  • The limit of f(x) is equal to the limits of the outside limits

The Squeeze Theorem cannot be applied if two-sided limit does not exist. In other words, if the right-hand and left-hand limits are not equal, the Squeeze Theorem will not work.

To solve with the Squeeze Theorem:

  • Make a double-sided inequality based on the nature of f(x)
  • Modify the inequality as needed
  • Solve the limits on both side of the inequality (they must be equal to apply the Squeeze Theorem)
  • The limit of f(x) is equal to the limits of the outside limits

Final The Squeeze Theorem Quiz

Question

Summarize the Squeeze Theorem in one sentence.

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Answer

The Squeeze Theorem is a limit evaluation method where we "squeeze" an indeterminate limit between two simpler ones; the "squeezed" function approaches the same limit as the other two functions surrounding it

Show question

Question

What is the step-by-step procedure for the Squeeze Theorem?

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Answer

  • Step 1: Create double-sided inequality based on the nature of f(x)
  • Step 2: Algebraically modify the inequality as needed
  • Step 3: Solve the limits on both sides of the inequality (they must be equal to continue)
  • Step 4: Apply the Squeeze Theorem

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Question

Should you always try the Squeeze Theorem first when solving limits?

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Answer

No! The Squeeze Theorem is a last resort method and only should be used if algebraic manipulation fails

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Question

What is another name for the Squeeze Theorem?

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Answer

Sandwich Theorem

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Question

Which family of functions are particularly good candidates for the Squeeze Theorem?

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Answer

trigonometric functions

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Question

What is the first method you should try for solving limits?

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Answer

You should always try algebraic or simple manipulation methods first.

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Question

Why is the Squeeze Theorem called the Squeeze Theorem?

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Answer

A function is "squeezed" between two simpler functions

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