### 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 .

• PROBLEM 2 : Compute .

• PROBLEM 3 : Compute .

• PROBLEM 4 : Compute .

• PROBLEM 5 : Compute .

• PROBLEM 6 : Compute .

• PROBLEM 7 : Compute .

• PROBLEM 8 : Assume that exists and . Find .

• 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 • PROBLEM 10 : Assume that Show that f is continuous at x=0 .