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Limit and Continuity (ÅÔÁÔμáÅФÇÒÁμèÍà¹×èͧ) Í. ¾ÑÅÅÀ ÎǺÊÁºÙóì [email protected] ÀÒ¤ÇÔªÒ¤³ÔμÈÒÊμÃìáÅÐʶÔμÔ ¤³ÐÇÔ·ÂÒÈÒÊμÃì ÁËÒÇÔ·ÂÒÅÑÂÁËÔ´Å SCMA 162 Cal. & Intro. to ODE. – p.1/30

LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit Ex ¤§¸Ò⁄ŁÒ¢˝§ Soln ¤—à¸ç„˙ŁÒ ä`Łä·Ø¶Ù¡„Ô´Ò`•ÕŁ¤Ø·

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Page 1: LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit Ex ¤§¸Ò⁄ŁÒ¢˝§ Soln ¤—à¸ç„˙ŁÒ ä`Łä·Ø¶Ù¡„Ô´Ò`•ÕŁ¤Ø·

Limit and Continuity

(ÅÔÁÔµáÅФÇÒÁµèÍà¹×èͧ)

Í. ¾ÑÅÅÀ ÎǺÊÁºÙóì[email protected]

ÀÒ¤ÇÔªÒ¤³ÔµÈÒʵÃìáÅÐʶԵԤ³ÐÇÔ·ÂÒÈÒʵÃì ÁËÒÇÔ·ÂÒÅÑÂÁËÔ´Å

SCMA 162 Cal. & Intro. to ODE. – p.1/30

Page 2: LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit Ex ¤§¸Ò⁄ŁÒ¢˝§ Soln ¤—à¸ç„˙ŁÒ ä`Łä·Ø¶Ù¡„Ô´Ò`•ÕŁ¤Ø·

Limit

Definition àÃÒ¨Ðà¢Õ¹ � � �� ��

� �� � � á·¹¡ÒáÅèÒÇÇèÒ

" ÅÔÁÔµ¢Í§ � ��

àÁ×èÍ � ÅÙèà¢éÒÊÙè ÁÕ¤èÒà·èҡѺ � "¶éÒàÃÒÊÒÁÒö·ÓãËé¤èҢͧ � ��

ÁÕ¤èÒã¡Åéà¤Õ§¡Ñº � â´Â¡Ò÷ÓãËé¤èÒ � ã¡Åéà¤Õ§¡Ñº áµè � � � Note ¨Ò¡¹ÔÂÒÁ¢éÒ§µé¹àÃÒÊÒÁÒöà¢Õ¹ � �� � � àÁ×èÍ � � á·¹ � � �

� ��� �� � �

àÃÒä´é¡Ó˹´ÇèÒ � � � «Öè§ËÁÒ¤ÇÒÁÇèÒ ã¹¡ÒÃËÒÅÔÁÔµ¢Í§¿Ñ§¡ìªÑè¹

� ��

àÁ×èÍ � ÅÙèà¢éÒÊÙè ¹Ñé¹ àÃҨРäÁè¾Ô¨ÒÃ³Ò ¡Ã³Õ·Õè � ÁÕ¤èÒà·èҡѺ

SCMA 162 Cal. & Intro. to ODE. – p.2/30

Page 3: LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit Ex ¤§¸Ò⁄ŁÒ¢˝§ Soln ¤—à¸ç„˙ŁÒ ä`Łä·Ø¶Ù¡„Ô´Ò`•ÕŁ¤Ø·

Limit

Ex ¨§ËÒ¤èҢͧ � � �� � �

� ��

� �

��

Soln ¨ÐàËç¹ÇèÒ � �� �� �

� �

�� äÁèä´é¶Ù¡¹ÔÂÒÁ·Õè¨Ø´ � � � áµèàÃÒ

ÊÒÁÒöËÒ¤èÒÅÔÁÔµ¢Í§ � ��

ä´é à¹×èͧ¨Ò¡ã¹¹ÔÂÒÁ¢Í§ÅÔÁÔµàÃÒ¾Ô¨ÒóÒ੾ÒФèҢͧ � ��

àÁ×èÍ � ÁÕ¤èÒ ã¡Åéà¤Õ§ � (áµè � � � �) à·èÒ¹Ñé¹

� � �� � �

� ��

� �

��

� � � �� � �

�� ��

�� ��

��� � �

� � � �� � �

�� � �

��

SCMA 162 Cal. & Intro. to ODE. – p.3/30

Page 4: LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit Ex ¤§¸Ò⁄ŁÒ¢˝§ Soln ¤—à¸ç„˙ŁÒ ä`Łä·Ø¶Ù¡„Ô´Ò`•ÕŁ¤Ø·

Limit

Ex ¨§ËÒ¤èҢͧ � � �� � �

� ��

àÁ×èÍ � �� ��

��

� ��

� �

�� ¶éÒ � � � �

� ¶éÒ � � �

Soln ã¹µÑÇÍÂèÒ§¹Õé � ��

¤ÅéÒ¡Ѻ � ��

ã¹µÑÇÍÂèÒ§·Õè¼èÒ¹ÁÒ áµè � ��

ä´é¶Ù¡¹ÔÂÒÁ·Õè � � � (¹Ñ蹤×Í � � � � �)

� � �� � �

� �� � � � �� � �

�� ��

�� ��

��� � � � � � �� � �

�� � � ��

ã¹µÑÇÍÂèÒ§¹Õé ¨ÐàËç¹ÇèÒ � � �� ��

� ��

äÁè¨Óà»ç¹µéͧà·èҡѺ � � �

SCMA 162 Cal. & Intro. to ODE. – p.4/30

Page 5: LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit Ex ¤§¸Ò⁄ŁÒ¢˝§ Soln ¤—à¸ç„˙ŁÒ ä`Łä·Ø¶Ù¡„Ô´Ò`•ÕŁ¤Ø·

Limit

Ex ¨§ËÒ¤èҢͧ � � �� � �

� ��

àÁ×èÍ � �� �� � � �

Soln ¨ÐàËç¹ÇèÒ � ��

äÁèä´é¶Ù¡¹ÔÂÒÁ·Õè̈ Ø´ � � � áµè¶éÒàÃÒ¾Ô¨ÒóҤèҢͧ � ��

àÁ×èÍ � ÁÕ¤èÒã¡Åéà¤Õ§¡Ñº �

� � � � � � � � � �

� ���

0.84147098

� ��

�0.99833417

� ��

0.95885108� ���

0.99958339

� ��

0.97354586

� ��� �

0.99998333

� ���

0.98506736

� ��� �

0.99999583

� ��

0.99334665

� ��� � �

0.99999983

¨ÐàËç¹ÇèÒàÁ×èÍ � ÁÕ¤èÒã¡Åéà¤Õ§ � áÅéÇ � ��

ÁÕ¤èÒ ã¡Åéà¤Õ§¡Ñº �

´Ñ§¹Ñé¹ àÃÒÊÃØ»ÇèÒ � � �� � �

� �� � � �

SCMA 162 Cal. & Intro. to ODE. – p.5/30

Page 6: LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit Ex ¤§¸Ò⁄ŁÒ¢˝§ Soln ¤—à¸ç„˙ŁÒ ä`Łä·Ø¶Ù¡„Ô´Ò`•ÕŁ¤Ø·

Limit

Ex ¾Ô¨ÒóҡÃÒ¿¢Í§ � �� � � � �

� ��

ÊÓËÃѺ � � �� ���

x

0.8

2

0.0

−0.4

0

−0.8

y

1.0

0.6

3

0.4

0.2

−0.21

−0.6

−1.0

−1−2−3

¨ÐàËç¹ÇèÒ � ��

ÁÕ¡ÒÃà»ÅÕè¹á»Å§¤èÒÃÐËÇèÒ§ �� áÅÐ � àÁ×èÍ � ÁÕ¤èÒ

ã¡Åéà¤Õ§ � ¨Ö§ÊÃØ»ÇèÒ � � �� � �

� ��

ËÒ¤èÒäÁèä´éËÃ×ÍäÁèÁÕ¤èÒ (do not exist)

SCMA 162 Cal. & Intro. to ODE. – p.6/30

Page 7: LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit Ex ¤§¸Ò⁄ŁÒ¢˝§ Soln ¤—à¸ç„˙ŁÒ ä`Łä·Ø¶Ù¡„Ô´Ò`•ÕŁ¤Ø·

Limit

Ex ¨§ËÒ¤èҢͧ � � �� � �

� ���

àÁ×èÍ � ��� ��

��

� ¶éÒ � � �

� ¶éÒ � � �

Soln ¨ÐàËç¹ÇèÒ àÁ×èÍ � � � áÅéÇ � ��� � � áµèàÁ×èÍ � � � áÅéÇ � ��� � �

�� � � �

� �

� ��

� ���

äÁèä´éÁÕ¤èÒà¾Õ§¤èÒà´ÕÂÇ àÁ×èÍ � ÁÕ¤èÒã¡Åéà¤Õ§¡Ñº �

´Ñ§¹Ñé¹ àÃÒÊÃØ»ÇèÒ � � �� � �

� ���

äÁèÁÕ¤èÒ

SCMA 162 Cal. & Intro. to ODE. – p.7/30

Page 8: LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit Ex ¤§¸Ò⁄ŁÒ¢˝§ Soln ¤—à¸ç„˙ŁÒ ä`Łä·Ø¶Ù¡„Ô´Ò`•ÕŁ¤Ø·

One-sided Limit (ÅÔÁÔµ·Ò§à´ÕÂÇ)

¨Ò¡µÑÇÍÂèÒ§·Õè¼èÒ¹ÁÒ¨ÐàËç¹ÇèÒ

�� � � �

� �

� ��

¤èҢͧ � ���

ÁÕ¤èÒà·èҡѺ � àÁ×èÍ � ÅÙèà¢éÒÊÙè � ·Ò§«éÒÂÁ×Í(¹Ñ蹤×Í ¾Ô¨ÒÃ³Ò ¡Ã³Õ·Õè � � �) àÃÒÊÒÁÒöà¢Õ¹ � � �

� � ��

� ��� � �

¤èҢͧ � ���

ÁÕ¤èÒà·èҡѺ � àÁ×èÍ � ÅÙèà¢éÒÊÙè � ·Ò§¢ÇÒÁ×Í(¹Ñ蹤×Í ¾Ô¨ÒÃ³Ò ¡Ã³Õ·Õè � � �) àÃÒÊÒÁÒöà¢Õ¹ � � �

� � � �

� ��� � �

SCMA 162 Cal. & Intro. to ODE. – p.8/30

Page 9: LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit Ex ¤§¸Ò⁄ŁÒ¢˝§ Soln ¤—à¸ç„˙ŁÒ ä`Łä·Ø¶Ù¡„Ô´Ò`•ÕŁ¤Ø·

One-sided Limit (ÅÔÁÔµ·Ò§à´ÕÂÇ)

Definition àÃÒ¨Ðà¢Õ¹ � � �

� �� �

� �� � � á·¹¡ÒáÅèÒÇÇèÒ

" ÅÔÁÔµ·Ò§«éÒ¢ͧ � ��

àÁ×èÍ � ÅÙèà¢éÒÊÙè ÁÕ¤èÒà·èҡѺ � "ËÃ×Í

" ÅÔÁÔµ¢Í§ � ��

àÁ×èÍ � ÅÙèà¢éÒÊÙè ·Ò§«éÒ ÁÕ¤èÒà·èҡѺ � "¶éÒàÃÒÊÒÁÒö·ÓãËé¤èҢͧ � ��

ÁÕ¤èÒã¡Åéà¤Õ§¡Ñº � â´Â¡Ò÷ÓãËé¤èÒ � ã¡Åéà¤Õ§¡Ñº áÅÐ � �

SCMA 162 Cal. & Intro. to ODE. – p.9/30

Page 10: LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit Ex ¤§¸Ò⁄ŁÒ¢˝§ Soln ¤—à¸ç„˙ŁÒ ä`Łä·Ø¶Ù¡„Ô´Ò`•ÕŁ¤Ø·

One-sided Limit (ÅÔÁÔµ·Ò§à´ÕÂÇ)

Definition àÃÒ¨Ðà¢Õ¹ � � �� �� �

� �� � � á·¹¡ÒáÅèÒÇÇèÒ

" ÅÔÁÔµ·Ò§¢ÇҢͧ � ��

àÁ×èÍ � ÅÙèà¢éÒÊÙè ÁÕ¤èÒà·èҡѺ � "ËÃ×Í

" ÅÔÁÔµ¢Í§ � ��

àÁ×èÍ � ÅÙèà¢éÒÊÙè ·Ò§¢ÇÒ ÁÕ¤èÒà·èҡѺ � "¶éÒàÃÒÊÒÁÒö·ÓãËé¤èҢͧ � ��

ÁÕ¤èÒã¡Åéà¤Õ§¡Ñº � â´Â¡Ò÷ÓãËé¤èÒ � ã¡Åéà¤Õ§¡Ñº áÅÐ � �

SCMA 162 Cal. & Intro. to ODE. – p.10/30

Page 11: LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit Ex ¤§¸Ò⁄ŁÒ¢˝§ Soln ¤—à¸ç„˙ŁÒ ä`Łä·Ø¶Ù¡„Ô´Ò`•ÕŁ¤Ø·

One-sided Limit (ÅÔÁÔµ·Ò§à´ÕÂÇ)

¨Ò¡¹ÔÂÒÁ¢Í§ÅÔÁÔµáÅйÔÂÒÁ¢Í§ÅÔÁÔµ·Ò§à´ÕÂÇ àÃÒ¨ÐàËç¹ÇèÒ

� � �� ��

� �� � � ¡çµèÍàÁ×èÍ � � �� �� �

� �� � � áÅÐ � � �� �� �

� �� � �

ã¹·Ò§¡ÅѺ¡Ñ¹

¶éÒ � � �� �� �

� �� � � � � �� �� �

� �� áÅéÇ � � �

� ��� ��

äÁèÁÕ¤èÒ

SCMA 162 Cal. & Intro. to ODE. – p.11/30

Page 12: LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit Ex ¤§¸Ò⁄ŁÒ¢˝§ Soln ¤—à¸ç„˙ŁÒ ä`Łä·Ø¶Ù¡„Ô´Ò`•ÕŁ¤Ø·

Limit (ÅÔÁÔµ)

Ex ¡Ó˹´¡ÃÒ¿¢Í§¿Ñ§¡ìªÑè¹ � ��

¨§ËÒ � � �� � �

� ��

áÅÐ � � �� �

� ��

Soln ����

��

� �

� � � � � ���

��� � � � � � � � �

� � � �

� �� � � � � � � �� � � �

� �� � �

� � �� � �

� ��

äÁèÁÕ¤èÒ� � � �

� � �� �� � � � � � �

� � �� ��

� � �� �

� �� � � �

ã¹µÑÇÍÂèÒ§¹Õé ÊѧࡵÇèÒ � � � � � � � � �� �

� �� � �

SCMA 162 Cal. & Intro. to ODE. – p.12/30

Page 13: LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit Ex ¤§¸Ò⁄ŁÒ¢˝§ Soln ¤—à¸ç„˙ŁÒ ä`Łä·Ø¶Ù¡„Ô´Ò`•ÕŁ¤Ø·

Infinity Limit (ÅÔÁԵ͹ѹµì)

Ex ¨§ËÒ¤èҢͧ � � �� � �

�� � (¶éÒÅÔÁÔµÁÕ¤èÒ)

Soln ¾Ô¨ÒóҡÃÒ¿¢Í§ � ��

� �

��

� �

�� �� �

àÁ×èÍ � ÁÕ¤èÒà¢éÒã¡Åé � áÅéÇ ��

ÁÕ¤èÒà¢éÒã¡Åé �

´Ñ§¹Ñé¹

�� � ¨ÐÁÕ¤èÒà»ç¹Í¹Ñ¹µì àÁ×èÍ � ÁÕ¤èÒà¢éÒã¡Åé �

¨Ò¡¡ÃÒ¿ ¨ÐàËç¹ÇèÒ àÁ×èÍ � ÁÕ¤èÒà¢éÒã¡Åé �

¡ÃÒ¿¢Í§ � ��

� � äÁèÅÙèà¢éÒÊÙè¨Ó¹Ç¹¨ÃÔ§ � ã´æ

( � äÁèãªè¨Ó¹Ç¹¨ÃÔ§ ) � � �� � �

�� � äÁèÁÕ¤èÒ

áµè㹡óչÕé àÃÒ¨Ðà¢Õ¹ÇèÒ � � �� � �

�� � � � �

SCMA 162 Cal. & Intro. to ODE. – p.13/30

Page 14: LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit Ex ¤§¸Ò⁄ŁÒ¢˝§ Soln ¤—à¸ç„˙ŁÒ ä`Łä·Ø¶Ù¡„Ô´Ò`•ÕŁ¤Ø·

Infinity Limit (ÅÔÁԵ͹ѹµì)

àÃÒà¢Õ¹ � � �� ��

� �� � � à¾×èͺ觺͡ÇèÒ ÅÔÁÔµ¢Í§ � ��

äÁèÁÕ¤èÒ ã¹á§è·Õè

� ��

ÁÕ¤èÒÁÒ¡æ àÁ×èÍàÃÒ·ÓãËé¤èÒ � ÅÙèà¢éÒÊÙè

�� � � �

� �

� ��

�� �� � �

� ��

do not exist

�� �� � �

� �� � �

�� � � �

� �

� � � � � �� �

�� �� � �

� ��

do not exist

�� �� � �

� �� � �

SCMA 162 Cal. & Intro. to ODE. – p.14/30

Page 15: LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit Ex ¤§¸Ò⁄ŁÒ¢˝§ Soln ¤—à¸ç„˙ŁÒ ä`Łä·Ø¶Ù¡„Ô´Ò`•ÕŁ¤Ø·

Infinity Limit (ÅÔÁԵ͹ѹµì)

Definition ¶éÒ � ��

ä´é¶Ù¡¹ÔÂÒÁ·Ñé§Êͧ¢éÒ§¢Í§ ( áµèÍÒ¨¨ÐäÁèä´é¶Ù¡¹ÔÂÒÁ·Õè¨Ø´ ¡çä´é ) áÅéÇ

� � �� ��

� �� � �

¨ÐËÁÒ¤ÇÒÁÇèÒ àÃÒÊÒÁÒö·ÓãËé¤èҢͧ � �� ÁÕ¤èÒÁÒ¡à·èÒã´¡çä´é

µÒÁ·ÕèàÃÒµéͧ¡Òà â´Â¡Òà ·ÓãËé¤èÒ � ã¡Åéà¤Õ§¡Ñº áµè � � �

Note:¹ÔÂÒÁ¢Í§ �� �

� ��� �� � � � ¤ÅéÒ¡Ѻ¹ÔÂÒÁ¢éÒ§µé¹

¹ÔÂÒÁ¢Í§ �� �

� �� �� �� � � �

�� �

� �� �� �� � � � �

�� �

� �� �� �� � �

áÅÐ �� �

� �� �� �� � � � ¤ÅéÒ¡Ѻ¹ÔÂÒÁ¢Í§ÅÔÁÔµ·Ò§à´ÕÂÇ

SCMA 162 Cal. & Intro. to ODE. – p.15/30

Page 16: LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit Ex ¤§¸Ò⁄ŁÒ¢˝§ Soln ¤—à¸ç„˙ŁÒ ä`Łä·Ø¶Ù¡„Ô´Ò`•ÕŁ¤Ø·

Infinity Limit (ÅÔÁԵ͹ѹµì)

Definition àÊ鹵ç � � ¨Ð¶Ù¡àÃÕ¡ÇèÒ àÊ鹡ӡѺá¹Ç´Ôè§ (Vertical

Asymtote) ¢Í§àÊé¹â¤é§ � � � ��

¶éÒ¢éÍã´ã¹µèÍ仹ÕéÍÂèÒ§¹éÍÂ˹Ö觢éÍà»ç¹¨ÃÔ§

� � �� ��

� �� � ��

� � �� �� �

� �� � ��

� � �� �� �

� �� � �

� � �� ��

� �� � � ��

� � �� �� �

� �� � � ��

� � �� �� �

� �� � � �

SCMA 162 Cal. & Intro. to ODE. – p.16/30

Page 17: LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit Ex ¤§¸Ò⁄ŁÒ¢˝§ Soln ¤—à¸ç„˙ŁÒ ä`Łä·Ø¶Ù¡„Ô´Ò`•ÕŁ¤Ø·

Infinity Limit (ÅÔÁԵ͹ѹµì)

Ex ¨§ËÒ¤èҢͧ � � �� � � �

� �� �

� áÅÐ � � �� � ��

� �� �

�Soln ¾Ô¨ÒóҡÃÒ¿¢Í§ � �

� �� �

� ¨ÐàËç¹ÇèÒ

��

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� � � �

àÊ鹵ç � � � ¤×Í àÊ鹡ӡѺá¹Ç´Ôè§

¢Í§àÊé¹â¤é§ � �

� �� �

� �

SCMA 162 Cal. & Intro. to ODE. – p.17/30

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Calculating Limits

Thm ¡Ó˹´ � à»ç¹¤èÒ¤§·Õè � � �� ��

� ��

áÅÐ � � �� ��

� ��

ËÒ¤èÒä´é áÅéÇ1.

� � �� ��

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2.

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3.

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4.

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5.

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� � �� ��

� ��

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� �� ¶éÒ � � �� ��

� �� � � �

SCMA 162 Cal. & Intro. to ODE. – p.18/30

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Calculating Limits

Thm (µèÍ)6.

� � �� ��

� � �� � � �� � � �

� ��� �� � �

àÁ×èÍ � à»ç¹¨Ó¹Ç¹àµçÁºÇ¡

7.

� � �� ��

� � � áÅÐ � � �� ��

� � 8.

� � �� ��

� � � � àÁ×èÍ � à»ç¹¨Ó¹Ç¹àµçÁºÇ¡

� � � � �

9.

� � �� ��

� �� � � � àÁ×èÍ � � � � �

àÁ×èÍ � à»ç¹àÅ¢¤Ùè áÅéÇ � �

. 10.

� � �� ��

����

� �� � � ���

� � �� ��

� ��

SCMA 162 Cal. & Intro. to ODE. – p.19/30

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Calculating Limits

Ex ¨§ËÒ¤èҢͧ � � �� �

� � � �

��� � � �

Soln

� � �� �

� � � �

��� � � � � � � �

� � � � � � �

�� � �

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��

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Note: ¨Ò¡µÑÇÍÂèÒ§·Õè¼èÒ¹ÁÒ ¨ÐàËç¹ÇèÒ

¶éҿѧ¡ìªÑè¹ � ��

à»ç¹¿Ñ§¡ìªÑè¹â¾ÅÔâ¹àÁÕ¹ËÃ×Ϳѧ¡ìªÑè¹àÈÉÊèǹáÅéÇ � � �

� ��� �� � � �

SCMA 162 Cal. & Intro. to ODE. – p.20/30

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Calculating Limits

Ex ¨§ËÒ¤èҢͧ � � �� � �

�� � � � � � �

��

��

Soln

�� �

� ��� �� � � �� �

� �

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SCMA 162 Cal. & Intro. to ODE. – p.21/30

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Calculating Limits

Ex ¨§ËÒ¤èҢͧ � � �� � �

�� � � �

��

� �

Soln

�� �� � �

�� � � � � �

� �

� �� �� � �

�� � � � � �

� � �

� � � � � � �

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�� � � � � �

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��

SCMA 162 Cal. & Intro. to ODE. – p.22/30

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Calculating Limits

Ex ¨§áÊ´§ÇèÒ � � �� � �

� � � � �

Soln

à¹×èͧ¨Ò¡

� � � ��

��

� ¶éÒ � � �

� � ¶éÒ � � �

� � �� � � �

� � � � � � �� � � �

� � � áÅÐ � � �� � ��

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� � � � � � � � �� � � �

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àÃÒÊÒÁÒöÊÃØ»ÇèÒ � � �� � �

� � � � � �

SCMA 162 Cal. & Intro. to ODE. – p.23/30

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Calculating Limits

Ex ¨§áÊ´§ÇèÒ � � �� � �

� � �� äÁèÁÕ¤èÒ

Soln

� � � �� � � �

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��

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àÃÒÊÒÁÒöÊÃØ»ÇèÒ � � �� � �

� � �

äÁèÁÕ¤èÒ

SCMA 162 Cal. & Intro. to ODE. – p.24/30

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Calculating Limits

Thm ¶éÒ � �� �� � ��

àÁ×èÍ � ÍÂÙèã¡Åé¡Ñº �

ÍҨ¡àÇé¹·Õè � �

áÅÐÅÔÁÔµ¢Í§ � ��

áÅÐ � ��

µèÒ§ÁÕ¤èÒ (exist) àÁ×èÍ � ÅÙèà¢éÒÊÙè áÅéÇ

� � �� ��

� �� �� � � �� ��

� �� Thm (Squeeze Thm) ¶éÒ � �� �� � �� � � ��

àÁ×èÍ � ÍÂÙèã¡Åé¡Ñº

ÍҨ¡àÇé¹·Õè � � áÅÐ � � �� ��

� �� � � � �� ��

� �� � �

áÅéÇ � � �� ��

� �� � �

SCMA 162 Cal. & Intro. to ODE. – p.25/30

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Calculating Limits

Ex ¨§áÊ´§ÇèÒ � � �� � �

� �� � �

� ��

� �

Soln àÃÒäÁèÊÒÁÒöãªéÊٵà � � �� ��

� � �� � � �� � � � � �� ��

� �� �

� � �� ��

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ä´é à¹×èͧ¨Ò¡ � � �� � �

� � �

� ��

äÁèÁÕ¤èÒ áµèÊÒÁÒöãªé Squeeze Thm ä´é

��

�� � � �� �

� � �

� ��� � �� � �� �

� � � �

áµèàÃÒ·ÃÒºÇèÒ � � �� � �

�� �

� � � áÅÐ � � �� � �

� � � �

´Ñ§¹Ñé¹ ¨Ò¡·ÄÉ®Õ àÃÒÊÃØ»ä´éÇèÒ � � �� � �

� �� � �

� ��

� �

SCMA 162 Cal. & Intro. to ODE. – p.26/30

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Limits at Infinity

¾Ô¨ÒóҡÃÒ¿¢Í§¿Ñ§¡ìªÑè¹ � �� �� �

��

� � � �0

1

�� � ��

� � � �

�� �

¨ÐàËç¹ÇèÒ ¶éÒàÃÒà¾ÔèÁ¤èÒ � ¢Öé¹àÃ×èÍÂæ áÅéǤèҢͧ � ��

¨ÐÁÕ¤èÒã¡Åéà¤Õ§¡Ñº � 㹡óչÕé àÃÒÊÒÁÒöà¢Õ¹͸ԺÒ¾ĵԡÃÃÁ¢Í§ � ��

ä´éÇèÒ

� � �� ��

� �� � �

SCMA 162 Cal. & Intro. to ODE. – p.27/30

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Limits at Infinity

Definition ¶éÒ � à»ç¹¿Ñ§¡ìªÑè¹·Õè¶Ù¡¹ÔÂÒÁ㹺ҧªèǧ� �

� áÅéÇ

� � �� ��

� �� � �ËÁÒ¤ÇÒÁÇèÒ àÃÒÊÒÁÒö·ÓãËé¤èҢͧ � ��

ÁÕ¤èÒã¡Åéà¤Õ§¡Ñº �

â´Â¡Ò÷ÓãËé � ÁÕ¤èҺǡÁÒ¡æ

Definition ¶éÒ � à»ç¹¿Ñ§¡ìªÑè¹·Õè¶Ù¡¹ÔÂÒÁ㹺ҧªèǧ

�� ��

áÅéÇ

� � �

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ËÁÒ¤ÇÒÁÇèÒ àÃÒÊÒÁÒö·ÓãËé¤èҢͧ � ��

ÁÕ¤èÒã¡Åéà¤Õ§¡Ñº �

â´Â¡Ò÷ÓãËé � ÁÕ¤èҵԴźÁÒ¡æ

SCMA 162 Cal. & Intro. to ODE. – p.28/30

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Limits at Infinity

Definition àÊ鹵ç � � � ¨Ð¶Ù¡àÃÕ¡ÇèÒ àÊ鹡ӡѺá¹ÇµÑé§(Horizontal Asymtote) ¢Í§àÊé¹â¤é§ � � � ��

¶éÒ � � �� ��

� �� � � ËÃ×Í � � �

� � � �� �� � �

0

1

�� ��� � �

��

� � � �

àÊ鹡ӡѺá¹ÇµÑ駢ͧàÊé¹â¤é§ � �� �

��

� � � � ¤×Í àÊ鹵ç � � �

SCMA 162 Cal. & Intro. to ODE. – p.29/30