=

(This is true because the expression approaches and the expression *x* + 3 approaches as x approaches
. The next step follows from the following simple fact. If *A* is a positive quantity, then
= *A* . )

=

=

=

(You will learn later that the previous step is valid because of the continuity of the square root function.)

=

(Inside the square root sign lies an indeterminate form. Circumvent it by dividing each term by , the highest power of *x* inside the square root sign.)

=

=

=

(Each of the three expressions , , and
approaches 0 as *x* approaches .)

=

=

= .

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=

(This is true because the expression approaches and the expression *x* + 3 approaches as *x* approaches
. The next step follows from the following simple fact. If *A* is a negative quantity, then
= - *A* so that = - ( - *A* ) = *A* . Please make sure that you think about and understand this before proceeding. )

=

=

=

(You will learn later that the previous step is valid because of the continuity of the square root function.)

=

(Inside the square root sign lies an indeterminate form. Circumvent it by dividing each term by , the highest power of *x* inside the square root sign.)

=

=

=

(Each of the three expressions , , and
approaches 0 as *x* approaches .)

=

=

= .

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=

(You will learn later that the previous step is valid because of the continuity of the logarithm function. Note also that the expression leads to the indeterminate form . Circumvent it by dividing each term by , the highest power of *x* .)

=

=

=

(The term approaches 0 as *x* approaches .)

=

=

= 0 .

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=

(You will learn later that the previous step is valid because of the continuity of the cosine function.)

=

=

(The expression leads to the indeterminate form
. Circumvent it by dividing each term by , the highest power of *x* in the
expression.)

=

=

=

(Each of the terms and approaches 0 as
*x* approaches .)

=

=

= .

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(As *x* approaches each of the expressions and
approaches 0. The following steps explain why.)

=

=

=

=

= 0 .

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=

(Circumvent this indeterminate form by dividing each term in the expression by . Division by also works . You might want to try it both ways to convince yourself of this. Also, BEWARE of making one of the following common MISTAKES : = or \ = .)

=

=

=

(Since approaches 0 and
approaches as *x* approaches ,
we get the following resultant limit.)

=

= .

(Thus, the limit does not exist.)

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= `` '' truein truein (BEWARE of making the following common MISTAKE : = . Realize also that the form `` '' is an indeterminate one ! It is not equal to 1 ! Circumvent it in the following algebraic ways.)

=

=

(Factor out the term . If you have time, try factoring out the term to convince yourself that it DOESN'T seem to help !)

=

=

=

=

=

(The expressions and approach 0 as *x* approaches .)

=

= .

= 9 .

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Wed Apr 2 10:10:40 PST 1997