Requirements for Squeeze theorem: 1- The bottom of fraction is non-negative. 2- There should be a positive common term (with the same power) on top and on the bottom of the fraction! X2; y2; x4;:::) 3- The common term is multiplied by the other terms in the numerator! (x2y is good, but x2 y2 is not!) Solution of part (b) using Squeeze. Then, by the Squeeze Theorem, lim x!0 x2 cos 1 x2 = 0: Example 2. Find lim x!0 x2esin(1 x): As in the last example, the issue comes from the division by 0 in the trig term. Now the range of sine is also 1; 1, so 1 sin 1 x 1: Taking e raised to both sides of an inequality does not change the inequality, so e 1 esin(1 x) e1; 1.
LIMITS OF FUNCTIONS USING THE SQUEEZE PRINCIPLE
The following problems involve the algebraic computation of limits using the Squeeze Principle,which is given below.
SQUEEZE PRINCIPLE : Assume that functions f , g , and h satisfy
and
.
Then
.
(NOTE : The quantity A may be a finite number, , or . The quantitiy L may be a finite number, , or .)
The Squeeze Principle is used on limit problems where the usual algebraic methods (factoring, conjugation, algebraic manipulation, etc.) are not effective. However, it requires that you be able to ``squeeze' your problem in betweentwo other ``simpler' functions whose limits are easily computable and equal. The use of the Squeeze Principle requires accurate analysis, deft algebra skills, and careful use of inequalities.
- PROBLEM 1 : Compute .
Click HERE to see a detailed solution to problem 1.
- PROBLEM 2 : Compute .
Click HERE to see a detailed solution to problem 2.
- PROBLEM 3 : Compute .
Click HERE to see a detailed solution to problem 3.
- PROBLEM 4 : Compute .
Click HERE to see a detailed solution to problem 4.
- PROBLEM 5 : Compute .
Click HERE to see a detailed solution to problem 5.
- PROBLEM 6 : Compute .
Click HERE to see a detailed solution to problem 6.
- PROBLEM 7 : Compute .
Click HERE to see a detailed solution to problem 7.
- PROBLEM 8 : Assume that exists and . Find .
Click HERE to see a detailed solution to problem 8.
- PROBLEM 9 : Consider a circle of radius 1 centered at the origin and an angle of radians, , in the given diagram.
a.) By considering the areas of right triangle OAD, sector OAC, and right triangle OBC, conclude that
.
b.) Use part a.) and the Squeeze Principle to show that
Click HERE to see a detailed solution to problem 9.
- PROBLEM 10 : Assume that
Show that f is continuous at x=0 .
Click HERE to see a detailed solution to problem 10. Thing arena 3.
Click HERE to return to the original list of various types of calculus problems.
Your comments and suggestions are welcome. Please e-mail any correspondence to Duane Kouba by clicking on the following address :
Duane KoubaWed Oct 15 16:55:51 PDT 1997
In this video we will learn all about the Squeeze Theorem.
Jenn, Founder Calcworkshop®, 15+ Years Experience (Licensed & Certified Teacher)
We will begin by learning that the Squeeze Theorem, also known as the Pinching Theorem or the the Sandwich Theorem, is a rule dealing with the limit of an oscillating function.
We will then learn how to conform, or squeeze, a function by comparing it with other functions whose limits are known and easy to compute.
So why is it called the Squeeze or Sandwich Theorem? Because that’s exactly what we are going to do. We’re going to squeeze one function between two other ones!
Collins efis 85 manual. So imagine you’re hungry and you decide to make a Peanut-Butter Sandwich (substitute the peanut-butter for your spread of choice).
First you need to acquire the necessary ingredients: peanut-butter and two pieces of bread.
Now we’re ready to assemble the sandwich. So we start by slathering the peanut-butter onto one slice of bread (or if you’re like me, on both) and then press the two slices of bread together. Viola! The peanut-butter is trapped or sandwiched between the two slices of bread, which means the stickiness (trickiness) is contained.
Example of the Squeeze Theorem or Sandwich Theorem
Well guess what, you just did the Squeeze Theorem! Math is delicious, isn’t it!
Okay, so in all seriousness, the peanut-butter is behaving like our unwieldy oscillating function, and the two pieces of bread are like our two easy-to-compute functions that are squeezing/containing it.
By why do we need the Squeeze Theorem, when we have our Rules for Indeterminate Forms?
Great question!
Well, in accordance with UC Davis, the Squeeze Principle is used on limit problems where the usual algebraic methods, such as factoring, common denominators, conjugation, or other algebraic manipulation are not effective.
So, let’s see this tasty theorem in action and walk through four examples of how to use and verify the Squeeze Theorem to evaluate a limit.
Squeeze Theorem Video
Squeeze Theoremap Calculus Notes
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