**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
.
*SOLUTION 2 :* First note 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
.
*SOLUTION 3 :* First note 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 - 2*x* < 0. Now divide each component by 3 - 2*x*, reversing the inequalities and getting
,
or
.
Since
,
it follows from the Squeeze Principle that
.
*SOLUTION 4 :* Note that DOES NOT EXIST since values of oscillate between -1 and +1 as *x* approaches 0 from the left. However, this does NOT necessarily mean that does not exist ! ? #. Indeed, *x*^{3} < 0 and
for *x* < 0. Multiply each component by *x*^{3}, reversing the inequalities and getting
or
.
Since
,
it follows from the Squeeze Principle that
.
*SOLUTION 5 :* First note 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 *x*^{2}, we get
and
.
Then
=
=
=
= .
Similarly,
= .
Thus, it follows from the Squeeze Principle that
= (does not exist).
*SOLUTION 6 :* First note that
,
so that
,
,
and
.
Then
=
=
=
= 5 .
Similarly,
= 5 .
Thus, it follows from the Squeeze Principle that
= 5 .
*SOLUTION 7 :* First note 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 *x*^{2} + 1 and multiply by *x*^{2} , getting
.
Then
=
=
=
=
= 0 .
Similarly,
= 0 .
It follows from the Squeeze Principle that
= 0 .
*SOLUTION 8 :* Since
=
and
= ,
it follows from the Squeeze Principle that
,
that is,
.
Thus,
.
*SOLUTION 9 :* a.) First note that (See diagram below.)
area of triangle OAD < area of sector OAC < area of triangle OBC .
The area of triangle OAD is
(base) (height) .
The area of sector OAC is
(area of circle) .
The area of triangle OBC is
(base) (height) .
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
.
*SOLUTION 10 :* Recall that function *f* is continuous at *x*=0 if
i.) *f*(0) is defined ,
ii.) exists ,
and
iii.) .
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 .
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