* 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

.

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* 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

.

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* 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

.

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* 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

.

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* 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).

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* SOLUTION 6 :* First note that

,

so that

,

,

and

.

Then

=

=

=

= 5 .

Similarly,

= 5 .

Thus, it follows from the Squeeze Principle that

= 5 .

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* 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 .

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* SOLUTION 8 :* Since

=

and

= ,

it follows from the Squeeze Principle that

,

that is,

.

Thus,

.

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* 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

.

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* 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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