### 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 between two 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.

### 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 Kouba
Wed Oct 15 16:55:51 PDT 1997