SOLUTIONS TO DERIVATIVES USING THE LIMIT DEFINITION


SOLUTION 1 :

tex2html_wrap_inline338

tex2html_wrap_inline340

tex2html_wrap_inline342

(Algebraically and arithmetically simplify the expression in the numerator.)

tex2html_wrap_inline344

(The term tex2html_wrap_inline330 now divides out and the limit can be calculated.)

tex2html_wrap_inline348

tex2html_wrap_inline350 .

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SOLUTION 2 :

tex2html_wrap_inline338

tex2html_wrap_inline358

(Algebraically and arithmetically simplify the expression in the numerator.)

tex2html_wrap_inline360

tex2html_wrap_inline362

(Factor tex2html_wrap_inline330 from the expression in the numerator.)

tex2html_wrap_inline366

(The term tex2html_wrap_inline330 now divides out and the limit can be calculated.)

tex2html_wrap_inline370

tex2html_wrap_inline372 .

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SOLUTION 3 :

tex2html_wrap_inline338

tex2html_wrap_inline380

tex2html_wrap_inline382

(Eliminate the square root terms in the numerator of the expression by multiplying

by the conjugate of the numerator divided by itself.)

tex2html_wrap_inline384

(Recall that tex2html_wrap_inline386 )

tex2html_wrap_inline388

tex2html_wrap_inline390

(The term tex2html_wrap_inline330 now divides out and the limit can be calculated.)

tex2html_wrap_inline394

tex2html_wrap_inline396

tex2html_wrap_inline398 .

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SOLUTION 4 :

tex2html_wrap_inline338

tex2html_wrap_inline406

(Get a common denominator for the expression in the numerator. Recall that division by tex2html_wrap_inline330 is the same as multiplication by tex2html_wrap_inline410 . )

tex2html_wrap_inline412

(Algebraically and arithmetically simplify the expression in the numerator. It is important to note that the denominator of this expression should be left in factored form so that the term tex2html_wrap_inline330 can be easily eliminated later.)

tex2html_wrap_inline416

tex2html_wrap_inline418

(The term tex2html_wrap_inline330 now divides out and the limit can be calculated.)

tex2html_wrap_inline422

tex2html_wrap_inline424

tex2html_wrap_inline426 .

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SOLUTION 5 :

tex2html_wrap_inline338

tex2html_wrap_inline434

(At this point it may appear that multiplying by the conjugate of the numerator over

itself is a good next step. However, doing something else is a better idea.)

tex2html_wrap_inline436

(Note that A - B can be written as the difference of cubes , so that

tex2html_wrap_inline440 . This will help explain the next step.)

tex2html_wrap_inline442

tex2html_wrap_inline444

(Algebraically and arithmetically simplify the expression in the numerator.)

tex2html_wrap_inline446

tex2html_wrap_inline448

(The term tex2html_wrap_inline330 now divides out and the limit can be calculated.)

tex2html_wrap_inline452

tex2html_wrap_inline454

tex2html_wrap_inline456

tex2html_wrap_inline458 .

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SOLUTION 6 :

tex2html_wrap_inline338

tex2html_wrap_inline466

tex2html_wrap_inline468

(Recall a well-known trigonometry identity :

tex2html_wrap_inline470 .)

tex2html_wrap_inline472

tex2html_wrap_inline474

(Recall the following two well-known trigonometry limits :

tex2html_wrap_inline476 and tex2html_wrap_inline478 .)

tex2html_wrap_inline480

tex2html_wrap_inline482

tex2html_wrap_inline484

tex2html_wrap_inline486

tex2html_wrap_inline488 .

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Duane Kouba
Thu Aug 29 15:59:12 PDT 1996