2.1.7 The Squeeze Theorem Flashcards Preview

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Flashcards in 2.1.7 The Squeeze Theorem Deck (13):
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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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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.
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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.

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Use the Squeeze Theorem to find
lim x→0 ∣x^3∣sin5x.

0

5

Use the Squeeze Theorem to find
lim x→0+√x / sin2x+1.

0

6

Use the Squeeze Theorem to find
lim x→0− √−x e^cos(−π/x).

0

7

Use the Squeeze Theorem to find
lim x→0 (ex−x−1)cos^2 2/x+x+1.

1

8

Use the Squeeze Theorem to find
lim x→1 |x−1|sin^2x+1.

1

9

Use the Squeeze Theorem to find
lim x→1− (1/x−√x) sin3/x−1 + √x.

1

10

If 3 ≤ f(x )≤ x^2+4x+7 for all x, find lim x→−2 f(x).

3

11

If −x^2+x+1 ≤ f(x) ≤−x+2 for all x, find lim x→1 f(x).

1

12

lim x→0 |x| sin10πx.

0

13

lim x→0√x^3+x^2 cosπ2x.

0

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