Approximate the limit of a function at a point by comparing it to another function whose limit at that point is known. Visualize the Squeeze Theorem graphically. Find the exact limit of a function at a point by using the Squeeze Theorem.

Theorem: If a function f is less than or equal to another function g in a neighborhood of a point x = a (not necessarily at a), then the limit of f as x approaches a is less than or equal to the limit of gas x approaches a, provided the limits exist. The given function lies between 1/x2and −1/x2. The previous theorem may be used to determine an interval within the range of the function in which the limit must lie. From the graph, it looks like the limit of the function as x approaches 5 is very close to −1/25. What is known for certain is that the limit lies in the narrow range [−1/25, 1/25]. The Squeeze Theorem is sometimes referred to as the Sandwich Theorem because the function whose limit is sought is being sandwiched between two functions, one from above, and the other from below. If those two functions have the same limit, then the function in question that lies between them must have that limit as well. For example, it is known that Mary’s height is always sandwiched between Laura’s and Fred’s, and that Laura and Fred approach the same height as they get close to 14 years old. Thus, by the Squeeze Theorem, Mary’s height must also approach the same height as Fred’s and Mary’s as she gets close to 14 years old. 1

The Squeeze Theorem may be used to find a limit of a complicated function exactly. In this example, the function looks complicated, and isn’t even defined at x= 0. But it can be shown that the limit of the function exists and is equal to 0 as x approaches 0, by noticing that the function lies between the functions |x| and −|x|, and then applying the Squeeze Theorem. Since the limit of |x| as x approaches 0 is 0, and the limit of −|x| as x approaches 0 is also 0, the limit of the given, complicated function as xapproaches 0 must also be 0.

Advanced Placement /AP Calculus AB: 2.1.7 The Squeeze Theorem

AP Calculus AB: 2.1.7 The Squeeze Theorem

Advanced Placement13 CardsCreated 7 days ago

The Squeeze Theorem helps find the limit of a function by "squeezing" it between two simpler functions whose limits are known and equal. If the given function is always between those two near a point, then its limit at that point is the same.

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The Squeeze Theorem

Approximate the limit of a function at a point by comparing it to another function whose limit at that point is known.
Visualize the Squeeze Theorem graphically.
Find the exact limit of a function at a point by using the Squeeze Theorem.

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Term

Definition

The Squeeze Theorem

Approximate the limit of a function at a point by comparing it to another function whose limit at that point is known.
Visua...

theorem

Theorem: If a function f is less than or equal to another function g in a neighborhood of a point x = a (not necessarily at a), then the li...

note

The Squeeze Theorem may be used to find a limit of a complicated function exactly.
In this example, the function looks compl...

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Use the Squeeze Theorem to find

lim x→0 ∣x^3∣sin5x.

Use the Squeeze Theorem to find

lim x→0+√x / sin2x+1.

Use the Squeeze Theorem to find

lim x→0− √−x e^cos(−π/x).

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AP Calculus AB: 2.1.7 The Squeeze Theorem

Term	Definition
The Squeeze Theorem	Approximate the limit of a function at a point by comparing it to another function whose limit at that point is known. Visualize the Squeeze Theorem graphically. Find the exact limit of a function at a point by using the Squeeze Theorem.
theorem	Theorem: If a function f is less than or equal to another function g in a neighborhood of a point x = a (not necessarily at a), then the limit of f as x approaches a is less than or equal to the limit of gas x approaches a, provided the limits exist. The given function lies between 1/x2and −1/x2. The previous theorem may be used to determine an interval within the range of the function in which the limit must lie. From the graph, it looks like the limit of the function as x approaches 5 is very close to −1/25. What is known for certain is that the limit lies in the narrow range [−1/25, 1/25]. The Squeeze Theorem is sometimes referred to as the Sandwich Theorem because the function whose limit is sought is being sandwiched between two functions, one from above, and the other from below. If those two functions have the same limit, then the function in question that lies between them must have that limit as well. For example, it is known that Mary’s height is always sandwiched between Laura’s and Fred’s, and that Laura and Fred approach the same height as they get close to 14 years old. Thus, by the Squeeze Theorem, Mary’s height must also approach the same height as Fred’s and Mary’s as she gets close to 14 years old. 1
note	The Squeeze Theorem may be used to find a limit of a complicated function exactly. In this example, the function looks complicated, and isn’t even defined at x= 0. But it can be shown that the limit of the function exists and is equal to 0 as x approaches 0, by noticing that the function lies between the functions \|x\| and −\|x\|, and then applying the Squeeze Theorem. Since the limit of \|x\| as x approaches 0 is 0, and the limit of −\|x\| as x approaches 0 is also 0, the limit of the given, complicated function as xapproaches 0 must also be 0.
Use the Squeeze Theorem to find lim x→0 ∣x^3∣sin5x.	0
Use the Squeeze Theorem to find lim x→0+√x / sin2x+1.	0
Use the Squeeze Theorem to find lim x→0− √−x e^cos(−π/x).	0

Use the Squeeze Theorem to find lim x→1 \|x−1\|sin^2x+1.	0
Use the Squeeze Theorem to find lim x→1− (1/x−√x) sin3/x−1 + √x.	0
If 3 ≤ f(x )≤ x^2+4x+7 for all x, find lim x→−2 f(x).	3
If −x^2+x+1 ≤ f(x) ≤−x+2 for all x, find lim x→1 f(x).	1
lim x→0 \|x\| sin10πx.	0
lim x→0√x^3+x^2 cosπ2x.	0

AP Calculus AB: 2.1.7 The Squeeze Theorem

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