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Infinite Limits (Part 1) — Transcript

by Dion Moses Baris · 4,888 words · 952 segments · language en · Watch on YouTube

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0:03Good afternoon lahat.
0:08So after po ng topic natin sa
0:12limits of functions in indeterminate
0:16forms. Again nga po sinabi nating
0:19indeterminate forms functions in
0:21determinate form. Yun po yung nagiging
0:25z0
0:28po siya.
0:29Yan po yung indeterminate form. Mga
0:32functions na nagiging ganyan once
0:34i-substitute natin yung value of x.
0:41So this time ibang case naman po. So we
0:44have the infinite limits and limits at
0:46infinity. Next day na lang po itong
0:49limits at infinity. Unahin muna natin
0:51itong infinite limits. So basahin ko
0:54lang introduction.
0:55In this chapter we will give a more
0:58informative answer and examine some
1:00related questions regarding the
1:02existence of a limit. In particular,
1:05recall example number two. I think
1:08naibigay ko naman na ito sa inyo n
1:09nakaraan ano sa pinakaunang lesson pa
1:12natin.
1:17Ito yung itsura ng graph ni example
1:19number two. We see from the graph figure
1:221.1.3 3 that x approaches
1:26zer from the right. So ito po yung
1:28tinatukoy no? X laging kaliwa or kanan
1:31lang yan. Since sabi dito x approaches
1:33zer from the right yan
1:37from the right
1:391 / x increases without bound. Pataas
1:43siya paganyan.
1:47While x approaches zero from the left.
1:50Ayan may arrow naman na nandiyan na eh.
1:52From the left.
1:541 over x decreases without bound. When
1:57we say without bound, walang hangganan.
2:00Decreases pababa. Kanina increases
2:03without bound this one decreases without
2:06bound.
2:09So symbolic symbolically we write limit
2:13of 1 / x as x approaches 0 from the
2:18right equals posi infinity.
2:21Ito po yun siya. And limit of 1/ x as x
2:27approaches zer from the left is
2:31infinity.
2:33So pakita lang natin ha. Recall natin ng
2:36konti itong dalawang to. T's itong graph
2:39na ito.
2:43Again po naibigay na itong example. Sabi
2:46diyan eh. Discuss the limit of fx if it
2:49exist. Sabi nga limit of fx as x
2:53approaches 0 equ limit of 1/ x. Ito yung
2:57ating function. Si 1/ x or f / x = 1/ x.
3:04Ganyan yan siya no?
3:08Then pagrough nga natin, ito yung
3:10kanyang itsura.
3:12So may paganyan tapos may paganito. So
3:16ganyan yung itsura ng graph ni
3:19fx
3:20= 1/ x. Ayan naman yan siya
3:28' ba nga yun sa mga first lesson natin
3:30determining limits. Wala pa yung
3:32substitution, wala pa yung mga theorems.
3:35Doun tayo sa pinaka kumbaga mano-mano
3:38system yung listing method. Itong table
3:41system natin 'no? Oh, first step nga
3:44nating ginagawa. Ito yung approaches
3:46natin. Zero ' ba? Zero. Ano nasa kaliwa
3:50ni zero? - 1. Ano nasa kanan ni 0? 1 -1
3:55pos 1 -1 + 1. Nasa example naman na
4:01natin ito.
4:06So from -1 ilalapit natin siya kay zero
4:11from the left kasi - 1 nasa left ni zer.
4:14So from the right ilalapit natin siya to
4:17zero from the right. T's
4:20ico-compute-compute niyo lang naman yan
4:22'di ba? Halimbawa nga po pag
4:24kinalculator natin
4:291 / -1 ito muna yan. Ito sagot -1.
4:34Tapos 1 / -.5. Ito yung sagot. -2
4:401/-.1.
4:431. Ito yung sagot natin. - 10. Tama po
4:47ba?
4:49Sinusundan lang natin yung formula ano?
4:52Idiretso niyo na lang sa calculator. 1 /
4:55-0.01-1.
4:59Pag pinalitan niyo ng
5:01nega
5:030.001
5:05yung denominator
5:07- 1000.
5:09Palitan niyo yung x ng001
5:15- 10,000
5:19and obviously
5:21habang pinapalaki natin I mean
5:23pinapadami yung zero dadami lang din
5:25yung zero sa kabila example
5:28gawin natin dito na
5:31gawin nating zeros
5:35kasi ang sabi
5:40X approaches zero from the left.
5:43Tatandaan ha yung negative diyan na
5:45exponent ang basa ay from the left. X
5:48approaches zero. Paunti-unti lumalapit
5:50to zero from the left. So say for
5:53example
5:55damihan lang natin ng 0 point
5:590. Okay. Sama ka friend 0.00
6:061 8 zeros yan
6:12negative po yan ha pag kinalculator po
6:15natin yan 1 / by yung -8 zeros tapos 1
6:21lalabas po diyan ay 1 billion actually
6:261 and 9 zeros
6:30oh ganyan yan siya 1 2 3 4 5 6 7 8 9 po
6:41sa kabila same procedure naman din halos
6:44nire-review lang natin ito.
6:47Yung nasa right na zero. Sulat ko para
6:52'di ba ito ang ating approaches
6:54approaches zero. X approaches zero. From
6:57the right we have pos one. From the left
7:00we have - 1
7:04taas approaches zero from the left. So
7:06paliit ng paliit, palapit ng palapit to
7:08zero. Katulad nga ng -001
7:168 zeros. Dito naman sa kabila
7:20approaching zero from the right ha.
7:24Inumpisahan nga ng 1. Sinubstitute natin
7:291 / palitan yung x ng 1
7:35then 1 /5
7:39lalabas 2. Positive na yun na 1 /1
7:43lalabas ay 10. 1 /01
7:48lalabas po ay 100.
7:521/001
7:561000 ang lalabas t's gawin mo lang
7:58tatlong zero 1 1/ 1 1/001
8:04lalabas ay 10,000
8:06same logic ulit damihan lang natin ng
8:09zerwa
8:12dito
8:16gawin nating
8:26Gan ulit 8 zeros so 0.001
8:31001
8:35positive yan ha
8:40so dinivide natin diyan 1 div by
8:47this one00
8:51ulitin ko001
8:55kasing zero lalabas ulit 1 billion 'yun
9:05with 9 zeros 1 billion 'di ba? Ganyan.
9:14As you can observe ba either from the
9:17left or either
9:21from the left or from the right habang
9:24dumadami yung zeros
9:27yung value of fx o yung y value natin ay
9:30dumadami lang din ng dumadami ng zeros.
9:34Dagdagan mo ng zero yung y or yung fx.
9:37magdadagdag lang din siya ng zero. Dito
9:40po natin nakuha 'yung concept of
9:42infinity.
9:44Kaya po pag nagka-calculator po tayo ng
9:47halimbawa 1 / 0, lumalabas po sa
9:52calculator natin ay error.
9:57Kasi ang katotohanan niyan ang sagot po
9:59ay infinite. No, dito pa nga lang eh
10:03hindi pa siya totally zero. May one pa
10:06siya sa dulo. Pero inabot na siya ng
10:09billyon ' ba? - 1 billion from the left,
10:14positive 1 billion from the right. Hindi
10:17pa siya totally zero non. Dagdagan mo pa
10:20ng zero magiging trilyon, magiging
10:22zilon, gazilon, etc. So how much more
10:25daw if 1 is being divided to zero? Doun
10:30po nakuha yung concept ng infinity.
10:34Yan infinity kasi even calculator kahit
10:39yung mga hybrid na computers
10:42hindi po kayang arukin kung ano yung
10:44dulo ng infinity. Infinity nga kasi siya
10:47eh. And kung mapapansin niyo ba yung
10:50pinakasagot nga natin dito the limit
10:52does not exist. Kasi ang limit isa lang
10:54dapat.
10:58Meaning to say yung from the left tapos
10:59from the right natin dapat same. In this
11:02case parehas naman silang 1 billion kaya
11:05lang magkaiba sila ng sign. This one ay
11:081 billion. This one ay pos 1 billion.
11:13Or ang mangyayari po diyan pag
11:15tinuloy-tuloy natin
11:18as x approaches zero from the left ito
11:22po ay paative infinity. And this one
11:27x approaches zero from the right.
11:29Papunta siya sa positive infinity. So
11:32magkaiba ba negative infinity positive
11:35infinity
11:37nawala siya. Ayan.
11:42So lumalabas dito the limit does not
11:44exist kasi dapat same yung left and yung
11:47right. Magkaiba yung left, magkaiba yung
11:49right. So tuloy ko lang yung pabasa.
11:52Note that the symbol infinity yung na
11:55pahiga. Ayan naka-highlight naman does
11:58not mean a real number kasi wala nga
12:00siyang ano eh. I mean wala pong
12:03nakakaarok ng dulo ng infinity kasi
12:06infinity nga po yan eh. Only God knows
12:08no kahit si Einstein pa or si Yuklid
12:12kung sino man ang pinakamatalino na
12:13mathematician tanungin hindi nila alam
12:16'yung exact value niyan. Wala pang
12:18nakakaimbento ng computer. Ako
12:20makakapagtala ng pinakadulo niyan kasi
12:22infinity nga siya. Symbol infinity does
12:25not mean a real number. We just gave a
12:29symbolic representation of the limit L.
12:33So ito nga approaching from the right
12:35positive infinity. positive infinity
12:38kasi sabi nga as x approaches from the
12:41right the value of y rises or increases
12:47without bound pataas siya e tingan niyo
12:49yung graph paganyan siya walang
12:52katapusan
12:54dito naman as x approaches from the left
12:57the value of y decreases without bound
13:01pababa ayan ang nature ng graph
13:04ito naman pataas
13:08When we say that the limit does not
13:09exist, we are saying that there is no
13:12real number. Kasi nga si infinity hindi
13:14daw siya real number that the function
13:17values are approaching. So writing
13:20infinity as a limit of a function simply
13:22express a particular way in which the
13:25limit does not exist. No does not exist
13:29naman pala. Ipinoproblema pa natin to.
13:33Hindi na lang natin itutuloyung pabasa
13:35niyan. Pwede niyo naman kayo na lang
13:38magbasa niyan no sa screenshot niyo na
13:40lang po 'yan.
13:46Yan
13:56bigay lang po tayo ng mga ilang formula
13:57na pwede nating magamit. Sabi dito
14:04ano ba yan? Mali
14:07if m
14:12if n is
14:16even
14:18yan then
14:24we can follow the following formulas. So
14:26say for example
14:28lim
14:301 / x nan
14:35as x approaches
14:400
14:43equ
14:47posi infinity
14:50posi infinity
14:58Another formula if e ano to lim
15:07x- a
15:10n
15:13as x approaches a
15:16equals din daw po siya automatically to
15:19positive infinity
15:21Pakita lang po natin ha.
15:28Kanina po dito
15:32diyan sa limit of 1/ x as x approaches
15:35zer ' ba dalawa nga po iyan. We have the
15:38negative infinity and the positive
15:41infinity. Actually sa lesson natin to be
15:44honest dalawa lang po lagi yung sagot
15:46dito sa infinite limits.
15:49It's either positive infinity or
15:53negative infinity. Ito lang pamimilian
15:55natin
15:57no kasi infinite limits nga eh. Again
16:00walang nakakaarok kung ano po yung dulo
16:04nung limit na yan.
16:06Yan. In this example dalawa po yung
16:09sagot. We have the negative infinity and
16:11the positive infinity. Kasi kung
16:14papansinin natin 1 x si x po ang
16:18exponent niya ay ' ba siya ay x^ 1 and 1
16:22is add. Yan cautious tayo sa pagtingin
16:26ng exponent po dito.
16:29Ulitin ko. 1 / x. Si x ang exponent niya
16:32ay 1 and 1 is
16:35odd. Kaya naging dalawa.
16:38Mamaya iano na natin yan pag even. Ito
16:41kasi condition
16:43if x is even. Galing
16:50yan. If x is even then we can follow the
16:54following formulas. May follow may
16:57following pa. Automatic daw positive
16:59infinity iyung sagot. Basta yung n ay
17:02even. Ano ba yung n? Ito po yung
17:06exponent. Yan. Yan. Ano ba pag sinabi
17:09nating even? Ma even po ito po ang mga
17:12halimbawa
17:16o 2 4 6 8 dot dot dot to the positive
17:24infinity. Yung mga divisible by 2 po
17:27yan.
17:30Kaya halimbawa
17:32instead na 1 / x gawin ko siyang 1 / x².
17:36Yan.
17:39So kahit hindi na tayo mag-f mag from
17:41the right, automatic daw ang magiging
17:42sagot ay positive infinity.
17:48Ayan ha. O sige kalaruin ko na lang
17:50example.
17:57Example
17:591
18:02limit
18:05as x approaches zero yan gawin natin 1 /
18:12x²
18:15or kahit gawin pa natin yan
18:19le x approach is 0 1
18:24x
18:266. Yan.
18:29O pag ganyan daw yung mga itsurahan
18:30natin, basta yung exponent ni X ay even.
18:36Automatic, huwag ka ng mag-compute,
18:39huwag ka ng gumawa ng table. Huwag na po
18:42yan no kasi time consuming pa yan.
18:48Automatic po ang isasagot niyo na
18:53positive infinitive. Positive infinity.
18:56Ba sabi ko sa inyo hindi naman ganon
18:58kahirapan. Basta ganyan po itsura. Yung
19:01nasa taas ay 1.
19:08Ta's uulitin ko nga
19:11yung kanina lim
19:15x as x approaches 0. Kaya dalawa may
19:19positive negative kasi nga yung x natin
19:23may invisible exponent 1 and 1 is add
19:26number. Dito na siya papasok sa
19:28pangalawang example natin.
19:36Yan
19:38o. Sulat ko lang. If
19:43nung n po ay exponent po yan ha. is add
19:52then
19:56yan
19:58yung mga add natin ito po ang example
20:021 3 5 7 dot dot dot
20:10meaning to say add not divisible by 2
20:16Ito naman po yung mga susundin nating
20:17formula.
20:29Halimbawa
20:32lim
20:34as x approaches 0
20:381
20:40x n
20:42equ
20:45conditional po siya
20:51zero nakalimutan Yan.
20:59So ayan halimbawa nawawala ako. So limit
21:02of 1 / x n
21:08yan
21:09as x approaches to zer kabaliktaran
21:13naman ito ay magiging
21:16negative
21:17infinity.
21:32Okay.
21:40Baburahin ko lang. Medyo nalito si sir.
21:46Sa ganitong pangit ba naman nasulat eh.
21:49Sarili na lito pa. So ulitin ko
21:51statement. If n is add then
21:56sundin nga natin itong mga formula na
21:58'to.
22:05If n is add ha. If n is add. So dalawa
22:08pala 'yan.
22:10Kasi nga kanina ba sa dito sa 1 / x
22:15dalawa yung naging sagot.
22:17Meron from the left, merong from the
22:19right. Yan.
22:22Solim
22:251/ x
22:27n
22:29as x approaches
22:32zero
22:34from the rightan from the right
22:39ag from the right walang problema
22:43ay
22:46positive infinity
22:52Ito nga lang agag from the left
22:55x approaches zero from the left
23:01tsaka siya magiging negative
23:05infinity
23:10x approach zer from the from the left
23:12katulad nito this one this is a perfect
23:15Example
23:18dahil nga x approaches
23:21zero from the left
23:24ano yung mga nangyari sa kanya value
23:26niya negative.
23:29Kapag tinuloy-tuloy pa nga natin yan,
23:32magiging negative infinity. Ayun yan ang
23:36perfect example.
23:40Or pwede rin naman ito
23:47lim
23:48as x approaches
23:51a
23:53from the right
23:56tapos 1 over
24:02x - a n
24:07Equals
24:08posi infinity.
24:12Madali lang naman class pag ano eh pag
24:16approaches from the right. Tatanda niyo
24:18lagi pag from the right from the right
24:20positive yon.
24:24Positive positive
24:26yan pag ano to ha yung exponent natin.
24:30Exponent natin na i-add katulad niyan.
24:32Yan
24:34ag from the right from the right
24:36positive positive infinity parang
24:38katulad nito from the right x approaches
24:42zero from the right mga positive yung
24:44sagot positive infinity
24:50dito lang nagkakaproblema ag from the
24:52left
24:56lim as x approaches A
25:02from the left equals nega
25:06infinity. Yan.
25:13Gaganyan lang siya.
25:18O lang. Ag from the left.
25:21From the left. Basta tatandaan 'yung
25:24exponent ay add.
25:27Katulad nito, ang exponent ni x ay 1
25:31which is add. O tignan niyo yung from
25:33the left nga pang ilang ulit ko na
25:36negative infinity.
25:40Ayan lang yan siya. Diyan lang naman
25:42nagkakaroon ng case to case, basic case
25:45to case. Saan ang galing ni?
25:49Yan.
25:56Tiyagain niyo na lang yan.
25:59O, screenshot niyo na lang po 'yan baka
26:01magagamit niyo. I think sa formula niyo
26:03nandoon naman din 'yan.
26:08Bigyan natin ng example para mas ano,
26:11mas klaro
26:13talaga ba, sir? Mas klaro.
26:23Example two. Halimbawa nga
26:28ba lim
26:32o gawin nating
26:34x^ 5
26:38as x approaches
26:400. Yan.
26:45Again nga po yan ang itsura natin. Limit
26:49of 1 / x^ 5 as x approaches zero. Yan.
26:57Titigan muna natin unang-una the
27:00exponent.
27:02At dahil add ang exponent natin, dalawa
27:05po ang pwedeng mangyari.Akita
27:08natin ang dalawa ang pwedeng mangyari.
27:17So we have from the left and from the
27:19right
27:22natin siyang lim.
27:27As x approaches zer from the right
27:31of 1/ x 5
27:37equals
27:43automatic ito magiging positive
27:46infinity
27:51magiging positive infinity
27:55pag dito naman
28:00lim
28:03x 5 as x approaches 0 from the left
28:11from the left nga
28:16exponent natin ay 5 which is add ito nga
28:19yun siya ba
28:21anong sagot
28:24O nakalimutan. pasensya na
28:27dapat meron itong ko lang formula ko.
28:38Masyadong nag-shortcut si sir that ito
28:40ay may 1 /
28:43x n tapos equals
28:47and this one dapat ito ay may 1 /
28:52x - a
28:55n
28:57saka yung equals
29:09negative
29:11infinity.
29:17Ito na nga yung sinasabi natin. Pag
29:18negative infinity automatic ay - agroach
29:23is zero from the left ito po 'yun.
29:28The answer is nega infinity.
29:37For example,
29:40limit
29:421 /
29:44x^ 7 as x approaches
29:49zero from the right. Ayan.
29:54Ito kasi walang nakalagay na dito sa una
29:56natin.
30:00Hindi ko nilagyan kung from the left or
30:02from the right. So kaya dinalawa natin
30:05kinuha natin from the right then the
30:07from the left.
30:10In this case ini-specify
30:12from the right.
30:15So again nga from the right
30:18approaches zero from the right tapos ito
30:20ay
30:23odd automatic positive infinity.
30:27Basta huwag niyo na litin yung sarili
30:28niyo basta may plus diyan positive
30:31infinity na yan.
30:36And this one lim
30:40x
30:429 as x approaches 0 from the left
30:48add ang exponent natin
30:51tapos from the left so negative infinity
30:55wala na nga halos computecute to eample
31:02number 3.
31:15Papaano naman yung sa mga ganito? Ito
31:17naman yung gamitan natin. This one.
31:22T's ito. Yan.
31:39So example number three we have lim
31:45lim
31:48tapos Tapos
31:58ayos sumulat
32:001 over
32:04x- 2
32:11squareag
32:14pala nating gawing square gawin nating
32:16to 4
32:20as x approaches
32:282
32:37an as x approaches
32:43Mm.
32:50So again po paganyan tignan agad si
32:53exponent tanungin add o even.
32:58Makita niyo agad na even yan. Huwag na
33:00kayong magpatumpik-tumpik ng sagot.
33:03Huwag niyo ng tignan dito kung positive
33:05kung kung from the left from the right o
33:08kahit wala yang sulat. Ulitin ko ha.
33:11Kahit yan from the left mali. Kahit yan
33:14from the left from the right o walang
33:17sulat.
33:19Ibig sabihin ano yan? Two sided
33:21function. Two sided function two sided
33:23limit. Automatic na ang sagot natin ay
33:29automatic na positive infinity.
33:33' ba ganun lang?
33:36Basta titignan lang natin
33:39exponent tapos kung from the left from
33:41the right. In this case automatic na yan
33:45kasi ano to eh even
33:49pag ginawa nga nating add
33:53halimbawa dito lim
34:11quantity x - 2
34:145
34:17ay sumulat
34:19as x approaches 0
34:22ayan
34:24this case
34:27add number siya yung ating exponent so
34:30dalawa yan pwedeng positive pwedeng
34:33negative infinity
34:38dalawahin natin
34:50x approaches 0
34:52from the right 1 over
34:58x- 2 5
35:01positive infinity
35:04positive infinity basta from the right
35:08may may plus diyan may positive positive
35:12infinity
35:15then lim as x approaches
35:20zero from the left copy paste lang
35:24x - 2
35:285
35:30Oh yun na naman Julie Julet super Julet
35:34basta ito ay add number tapos ito ay
35:37negative add number negative automatic
35:43negative infinity
35:47kailan lang hindi magiging negative
35:49infinity ang sagot kapag si exponent
35:54natin ay even say for example naging
35:57four kaya nga ito four hindi na natin
36:01inalam yung Yung from the left, yung
36:03from the right, yung left hand limit,
36:06right hand limit hindi na kasi automatic
36:07siyang positive infinity.
36:22How about
36:33limit as x approaches 2 from the left
36:36same pain. Gawin natin siyang
36:401 /
36:44x - 2 quantity x - 2^ 3
36:49wala ng computecute yan halos eh. Ito
36:51yung kagandahan sa lesson na to. Walang
36:54computecute.
36:56Tingan lang si exponent
36:59add dahil add. Tingan po ito. From the
37:02left sagot negative infinity.
37:14Gawin nating from the right to from the
37:18right. So once again
37:21exponent ay 3. I-add number si 3 not
37:26divisible by 2. O baka mamaya lito pa
37:29tayo sa even at sa add. Add 135 79 etc
37:33etcera. even 2 4 6 8 10 etc etca o dahil
37:38nga add ta's ito ay from the right sagot
37:46positive
37:48infinity
37:50screenshot niyo na yan
37:59tapos ito nga po kinor ko na onong
38:01formula na to
38:06Yan.
38:10Kausin niyo na lang 'to. Yan.
38:13Nakalimutan kasi.
38:17Aayusin ko na lang para hindi
38:19nakakalito.
38:26X approaches A from the left. Yan.
38:34X approaches A from the left. Yan.
38:39Okay na.
38:48Again po ha, uulitin ko lang. Ginagawa
38:50natin to pag infinite limits pag
38:53nag-a-undefined.
38:55Yung unang lesson natin yung
38:56nagfa-factoring tayo. Special product
38:59rationalization kasi nag-0 over 0. Ibang
39:04kaso po si 1 / 0. Ito po ay hindi po
39:07indeterminate. Yan po ay undefined. Pag
39:11ang zero ay denominator, tawag natin ay
39:14undefined. Pag parehas numerator
39:17denominator ay zero in determinate.
39:19Ulitin ko.
39:23Galing
39:25in determinate
39:27undefined. Pwede ang palitan. 2 3 4
39:32basta zero yung nasa baba. undefined mat
39:36error 'yan kasi nga ang katotohanan
39:37infinity po ang sagot either positive or
39:40negative infinity pahirapan naman natin
39:43mga example syempre
39:47para magamit naman 'yung mga formula
39:49diyan sa photocopy.
39:53Example
39:58number four.
40:01Paano naman daw pag ano infinite limit
40:03of rational functions?
40:09Mabibigyan po ng halimbawa
40:17limit
40:23root of x squ square
40:30-9
40:36all over x- 3
40:42as x approaches 3 from the right
40:51as x approaches 3 from the
40:55O check natin
40:59pag ganito. Halimbawa sa quiz or sa exam
41:01hindi ko po yan sasabihin kung anong
41:03gagawin niyo. Kung magfa-factoring ba
41:05kayo, mag-special product o gagawin niyo
41:07itong lesson natin ngayon sa infinite
41:09limit. Dapat kayo maga-identify niyan
41:11kasi analysis nga yan e. Critical
41:14thinking. Kayo mag-identify niyan.
41:18Natignan niyo halimbawa pag ginawa
41:20natinung substitution agad
41:23example
41:28root of 3 s kasi x approaches 3 e
41:31palitan yung x ng 3
41:34- 9 all over
41:393 - 3
41:43yung x po ay pinalitan lang natin ng 3
41:46based po sa approach approaches natin to
41:48pag kinalculator po natin yan 3 s ay 9 -
41:539 0 ro 0 ay 0 tapos sa baba 3 - 3 0 s
42:00root of 0 alam niyo na yan zero ' ba 0
42:03/0 o nagkaroon tayo ng indeterminate 'di
42:07ba naging indeterminate form siya so pag
42:10indeterminate form ginagamitan nga natin
42:12ng factoring special product or
42:15rationalization
42:17Depende po
42:19no.
42:21Baburahin ko lang.
42:34An in this case sabi ko nung nakaraan,
42:37pag may square root, ang kadalasang
42:38ginagamit niyo rationalization.
42:42Yan
42:44ang kadalasang ginagamit ay
42:46rationalization
42:53pero hindi naman laging rationalization
42:55lang yung ginagamit
42:59pwede pa tayong gumamit ng other means.
43:08Halimbawa po in this case
43:12nagkakaproblema kasi sa baba nagiging
43:13zero po yan eh. Nagiging 3 - 3 equ 0. So
43:18focus tayo sa taas. Sot
43:22of the quantity x² - 9. Yan x² - 9.
43:28Mapapansin niyo x² - 9.
43:31Same din po yan sa Kokopyahin ko agad
43:34'yung link baka makalimutan.
43:37As x approaches 3
43:40yan from the right 3 po yan.
43:50Same po ito sa x s - 3 s kasi ang 9 3 s.
43:58Tama po ba?
44:01Soung 9 pinalitan lang natin ng 3 s.
44:11all over
44:13x - 3
44:19kasi indeterminate eh pwede nating
44:21gamitan ng factoring special product or
44:24rationalization
44:27in this case l ganyan itsura niya x s- 3
44:30s yan po ay difference of 2 squ ha
44:37difference minus kasi 2 squ parehas
44:39silang naka-square. So fa-factor out
44:41lang natin. Applying the difference of 2
44:43squares. So ganito po mangyayari.
44:46Konting review. Next week exam na. First
44:50term natin ay x.
44:53Second term natin ay 3
44:56positive negative 'di ba?
45:03T's huwag kalimutan yung square root.
45:08square root ba parang
45:11from this to thisctor
45:15out kasi natin
45:19all over
45:22x - 3
45:26copy lim as x approaches 3
45:31from the right
45:35kaya lang class hindi pa kayo pwedeng
45:36basta Basta-basta mag-cancel na lang
45:38yan. Ulitin ko ha. Hindi pwedeng
45:41basta-basta mag-cancel.
45:47Bakit hindi ba sabasta magka-cancel?
45:49Kasi nga
45:52may square root po ito. I mean itong mga
45:55factors dito ay nasa loob ng square root
45:58ng radical.
46:00Ka pwedeng basta ika-cancel mo yan kasi
46:02nga may radical sign. So gagawaan natin
46:06ng paraan. Ganun naman 'yun eh. Lahat ng
46:09bagay nagagawaan ng paraan.
46:12Ang problema kasi
46:14yung numerator may radical sign. Yung
46:16denominator may radical sign.
46:20And yung denominator lagyan din natin ng
46:22radical sign.
46:25Or again hindi tayo basta-basta
46:27maglalagay ng radical sign. Halimbawa x-
46:303
46:32tapos nilagyan natin siya. galing.
46:35Lagyan natin siya ng radical sign. Ba
46:37'yan?
46:39Radical sign na ganyan. May nagbago 'di
46:42ba? Chin mo siya. May naidagdag.
46:45So para ma-maintain natin yung
46:48originality niya, lagyan natin ng
46:54square.
46:59Lalagyan lang natin ng square
47:13kasi nasa taas.
47:21Eh
47:31nasa taas natin.
47:34Kaya sabi ko sa inyo dapat nire-recall
47:36natin yung anoung mga
47:40na-discuss before nung high school kayo.
47:43Radicals, mga rules sa radicals.
47:46Kasi ano po ito calculus eh. Dapat nga
47:49may mga cosine sin cosine tangent tayo.
47:51Hindi ko na lang isasama 'yun. This one
47:53pwede nating paghiwalayin 'yan.
47:56Pwede nating gawing
47:59root
48:00of x + 3
48:06s root
48:08of x- 3. Yan ba pwede paghiwalayin
48:15taas. Pag pinag-times mo 'yan,
48:20pinag-times mo po sila, pinag-times mo,
48:22babalik siya diyan.
48:25Pinaghiwalay lang po natin. Pwede po
48:27'yan sa radicals.
48:30O-copyan po si Lim
48:35as X approaches 3 from the right
48:39equals
48:41yan o. Pag ganyan na. Tingnan niyo na
48:43may ma 'di ba? Parang may maka-cancel na
48:46tayo.
48:48Mayon na po. Ang pwede na nating
48:50ma-cancel ito at ito.
48:58Operate lang natin. Cancel po natin.
49:02Ruling po. Pag nagdi-divide tayo sa
49:04algebra.
49:07This factor it has an exponent one
49:11invisible exponent 1. Ba ag nagdi-divide
49:15tayo involving exponents
49:19pagdating sa algebra na may involved na
49:21exponent ang ginagawa lang natin
49:24minus
49:29mangyayari dito cancel po ito. Cancel
49:32to. Matitira na lang one.
49:38Sulitin ko ha. Ma-cancel po ito.
49:41T's cancel ito. Magiging one na lang.
49:47So kokopy mo yung natira which is this
49:50one.
49:56square square root of the sum of x + 3
50:01all over
50:04square root
50:17of x - 3.
50:23Kam magtatanong ha bakit naging ganyan
50:25na lang. So from this
50:29to this T's nawala po ito.
50:32Ire-write ko lang dito.
50:39Over ganyan.
50:43Ire-write ko lang.
50:50Parang ganyan yan kasi siya
50:53this one
50:56dahil nga same silang ro x - 3
51:02taas ito may invisible exponent 1
51:06cancel po ito tapos cancel yung 2
51:09magiging one parang ano 'yan eh magiging
51:11ganito siya ' ba ang ruling diyan pag
51:13nagdi-divide minus ng exponent so ip
51:16Pakita lang natin step by step. 1 - 2
51:19magiging - 1.
51:27Magiging - 1 ganyan. Kasi nga 1 - 2 =
51:33-1.
51:35At pag may exponent po tayo -1,
51:38ibinababa po natin yan. magiging 1 /
51:42ro x - 3.
51:45Kaya po
51:47naging ano na lang siya dito
51:51root x - 3 ta's yung natira
51:56yung natira po na ro x + 3 kinopy-paste
52:00lang natin diyan.
52:07na lang makakagulo pa
52:23tapos syempre kopyahin po natin yung
52:26huwag niyong kakalimutan
52:30lim aensa
52:33asx approaches three from the
52:48Then after niyan yung ganyan wala na
52:50tayong magagawa diyan. Hindi na siya
52:52factorable
52:54wala na ring pwedeng special product na
52:56gawin. Yung rationalization hindi rin
52:59natin maa-apply.
53:02Actually yung rationalization magagamit
53:04natin pa ganito yung itsura. X +
53:083. Yan po yung pwede.
53:13Yan. Yan yung pwede nating gamitin kasi
53:15pwede natin siyang multiplyan ng pwede
53:18natin siyang i-multiply sa
53:22ro x - 3
53:26para maging
53:28ba maging ganyan. s - 3 s para cancel
53:33cancel
53:35pwede sana yan kaya lang ang problema
53:38yung square root included si + 3 so
53:42hindi siya ganito yung itsura niya
53:44included si+ 3 nasa loob ng square root
53:46so yung rationalization hindi natin
53:48magagamit
53:51in that case
53:53wala ng possible na factoring special
53:55product or rationalization
53:59no choice ka. Substitute ka na.
54:02Okay. Substitute.
54:06Substitute natin yung x papalitan lang
54:09natin ng 3.
54:15So magiging 3 + 3 all over. Ito wala
54:19talagang magagawa. 0 - 3 3 + 3 ito kahit
54:24papaano magiging ro pa all over
54:29mali ako sorry I mean itong denominator
54:32yung mali excited si sir eh
54:36zero kasi yung sagot niyan yan e 3 - 3
54:40magiging zero nga yan
54:44alam niyo na
54:47ro 0 is 0 ta's nasa taas ay sare ro 6.
54:50Ito nga yung sinasabi nating class na
54:52undefined no? Hindi siya indeterminate
54:55na nagiging 0/0 lang. O nag-factoring na
55:00tayo kung ano-ano ng ginawa na natin
55:02diyan. Ending.
55:06Ano pala siya? Undefined yung zero niya
55:10naging
55:13ay I mean yung denominator niya naging
55:15zero. So pag ganyan nga undefined
55:18infinite limit po yan. So either
55:21positive or negative
55:24infinity 'yung sagot natin.
55:29Dito natin 'yan siya titingnan. Yan kung
55:32from the left, from the right. ' Ba po
55:34may formula diyan sa inyo. Tama ' ba? O
55:38tingnan niyo habang nakikinig kayo nito,
55:41dapat may formula kayo. Syempre mahirap
55:43mangulan ng formula. Ako nga may
55:45tinitignan na libro.
55:51Yan.
55:52Ang pinakasimple dito ay
55:55may mas complex pa kasi na formula.
55:58Hindi ko na sinama yan. So ito na lang
56:01yung gagamitin nating formula.
56:03No, sabi dito
56:05if the numerator is constant
56:11constant
56:13tapos sa baba ay zero
56:18from the right ang sagot natin ay
56:21positive
56:29infinity
56:31kabaliktaran
56:33constant over
56:370 from the left negative infinity.
56:44Yan
56:46yung letter C pala ay constant ha.
56:49Tinuro ko naman ano constant o katulad
56:51nito yung sare root of 6 constant yan '
56:55ba sinabi nating constant number. Yan
56:59yung letter C. Any
57:02bahala na kung anong lumabas doon. Basta
57:04yan yung tinutukoy na C constant yan
57:07yung C tos yung nasa baba natin ay zero.
57:10So yan kung positive siya or negative.
57:15I mean kung from the left or from the
57:17right.
57:21And this one yung constant natin ay
57:23negative.
57:28constant natin ay negative
57:30tapos yung zero natin ay from the right
57:35negative infinity yan
57:39then i-correct niyo po sa handouts niyo
57:41kasi mali yung nasa handout
57:45yung constant ay negative and the zero
57:49approach ano to zer from the right from
57:52the left rather
57:56Nega infinity
58:00parang ano simpleng ano lang yan eh ba
58:03yung rule sa division
58:05positive divided by positive positive
58:08positive div negative negative neg div
58:11negative negative neg div by negative o
58:15ito na naman malinis sir positive
58:19yan kasi nasa handout niyo nga
58:22nakaagative
58:24nakegative dito. Paki-correct yan sa
58:26handout niyo. Hindi po yan negative.
58:28Dapat ito po ay positive kasi negative
58:31div negative ay positive. In this case.
58:35So ano magiging sagot natin dito? Yung s
58:38root of 6 natin walang sign sa unahan.
58:40Yan ay automatic positive. Tapos itong
58:43zero
58:45doon mo lang siya titignan. Ayan oh.
58:48X approaches 3 from the right. E ito po
58:51ay lagyan niyo lang dito
58:55zero from the right. O titingnan niyo
58:58diyan o constant na positive tapos zero
59:02from the right. Ito yung sagot ay
59:04positive infinity.
59:08O ganyan lang yun siya. Final answer.
59:13Contination na lang tayo ng discussion
59:16nito sa face to face class natin.

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