### 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 :

kouba@math.ucdavis.edu

*Duane Kouba *

Wed Oct 15 16:55:51 PDT 1997