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.

SOLUTIONS TO LIMITS USING THE SQUEEZE PRINCIPLE
SOLUTION 1 : First note that

because of the well-known properties of the sine function. Since we are computing the limit as x goes to infinity, it is reasonable to assume that x > 0 . Thus,

Since

it follows from the Squeeze Principle that

because of the well-known properties of the cosine function. Now multiply by -1, reversing the inequalities and getting

or

Next, add 2 to each component to get

Since we are computing the limit as x goes to infinity, it is reasonable to assume that x + 3 > 0. Thus,

Since

it follows from the Squeeze Principle that

because of the well-known properties of the cosine function, and therefore

Since we are computing the limit as x goes to infinity, it is reasonable to assume that 3 - 2x < 0. Now divide each component by 3 - 2x, reversing the inequalities and getting

or

Since

it follows from the Squeeze Principle that

for x < 0. Multiply each component by x³, reversing the inequalities and getting

or

Since

it follows from the Squeeze Principle that

so that

and

Since we are computing the limit as x goes to infinity, it is reasonable to assume that x+100 > 0. Thus, dividing by x+100 and multiplying by x², we get

and

Then

=

=

= .

Similarly,

= .

Thus, it follows from the Squeeze Principle that

so that

and

Then

=

=

=

= 5 .

Similarly,

Thus, it follows from the Squeeze Principle that

and

so that

and

Since we are computing the limit as x goes to negative infinity, it is reasonable to assume that x-3 < 0. Thus, dividing by x-3, we get

or

Now divide by x² + 1 and multiply by x² , getting

Then

=

=

=

=

= 0 .

It follows from the Squeeze Principle that

and

it follows from the Squeeze Principle that

that is,

Thus,

area of triangle OAD < area of sector OAC < area of triangle OBC .

The area of sector OAC is

The area of triangle OBC is

It follows that

or

b.) If , then and , so that dividing by results in

Taking reciprocals of these positive quantities gives

or

Since

it follows from the Squeeze Principle that

i.) f(0) is defined ,

ii.) exists ,

and

First note that it is given that

i.) f(0) = 0 .

Use the Squeeze Principle to compute . For we know that

so that

Since

it follows from the Squeeze Principle that

ii.) .

Finally,

iii.) ,

confirming that function f is continuous at x=0 .