LimitandContinuity - mucc.mahidol.ac.thmucc.mahidol.ac.th/~scphc/Classes/scma115/LIMIT.pdf · Limit...
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Infinity Limit (ÅÔÁԵ͹ѹµì)
Ex ¨§ËÒ¤èҢͧ � � �� � � �
� �� �
� áÅÐ � � �� � ��
� �� �
�Soln ¾Ô¨ÒóҡÃÒ¿¢Í§ � �
� �� �
� ¨ÐàËç¹ÇèÒ
��
� �
�
�� � �� �
�
� � �
� � �� � � �
� �� �
� � �
� � �� � ��
� �� �
� � � �
àÊ鹵ç � � � ¤×Í àÊ鹡ӡѺá¹Ç´Ôè§
¢Í§àÊé¹â¤é§ � �
� �� �
� �
SCMA 162 Cal. & Intro. to ODE. – p.17/30
Calculating Limits
Thm ¡Ó˹´ � à»ç¹¤èÒ¤§·Õè � � �� ��
� ��
áÅÐ � � �� ��
� ��
ËÒ¤èÒä´é áÅéÇ1.
� � �� ��
� � �� � � �� � � � � �� ��
� �� � � � �� ��
� ��
2.
� � �� ��
� � �� � � �� � � � � �� ��
� �� �
� � �� ��
� ��
3.
� � �� ��
����
� �� � � ���
� � �� ��
� ��
4.
� � �� ��
� � �� � � �� � � � � �� ��
� �� �
� � �� ��
� ��
5.
� � �� ��
� �� � �� �
� � �� ��
� ��
� � �� ��
� �� ¶éÒ � � �� ��
� �� � � �
SCMA 162 Cal. & Intro. to ODE. – p.18/30
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
Calculating Limits
Ex ¨§ËÒ¤èҢͧ � � �� �
� � � �
��� � � �
Soln
� � �� �
� � � �
��� � � � � � � �
� � � � � � �
�� � �
� � � �� � � � � �
� � �
� � � � �� �
�� � ��
� � � �� �
� � � � �� �
�
� ��
�
��
�
� � � �� �
Note: ¨Ò¡µÑÇÍÂèÒ§·Õè¼èÒ¹ÁÒ ¨ÐàËç¹ÇèÒ
¶éҿѧ¡ìªÑè¹ � ��
à»ç¹¿Ñ§¡ìªÑè¹â¾ÅÔâ¹àÁÕ¹ËÃ×Ϳѧ¡ìªÑè¹àÈÉÊèǹáÅéÇ � � �
� ��� �� � � �
SCMA 162 Cal. & Intro. to ODE. – p.20/30
Calculating Limits
Ex ¨§ËÒ¤èҢͧ � � �� � �
�� � � � � � �
��
�
��
�
Soln
�� �
� ��� �� � � �� �
� �
� � ��
�
�� �
� ��� ��� � � �� �
� �
�� �
� ��� �� � � ��
�
�� �
� ��� �� � � � �� �
� ��� �� �
�
�� �
� ��� ��
�� �
� ��� �� � � �� �
� ��� ��
�
�� � � � � � � � � �
� � � � � �
� ��
� �
SCMA 162 Cal. & Intro. to ODE. – p.21/30
Calculating Limits
Ex ¨§ËÒ¤èҢͧ � � �� � �
�
�� � � �
��
� �
�
Soln
�� �� � �
�
�� � � � � �
� �
�
� �� �� � �
�
�� � � � � �
� � �
� � � � � � �
� � � � � � ��
� �� �� � �
� � � � � � �
� � � � � � � � � �
� �� �� � �
� �
� � � � � � � � � � � �� �� � �
�
�� � � � � �
�
�
�� �� � �
� � � � � � �
�
��
SCMA 162 Cal. & Intro. to ODE. – p.22/30
Calculating Limits
Ex ¨§áÊ´§ÇèÒ � � �� � �
� � � � �
Soln
à¹×èͧ¨Ò¡
� � � ��
��
� ¶éÒ � � �
� � ¶éÒ � � �
� � �� � � �
� � � � � � �� � � �
� � � áÅÐ � � �� � ��
� � � � � � �� � ��
�� � � �
� � � �� � ��
� � � � � � � � �� � � �
� � �
àÃÒÊÒÁÒöÊÃØ»ÇèÒ � � �� � �
� � � � � �
SCMA 162 Cal. & Intro. to ODE. – p.23/30
Calculating Limits
Ex ¨§áÊ´§ÇèÒ � � �� � �
� � �� äÁèÁÕ¤èÒ
Soln
� � � �� � � �
� � ��
� � � �� � � �
��
� � � � � � �� � ��
� � ��
� � � �� � ��
� ��
� ��
àÃÒÊÒÁÒöÊÃØ»ÇèÒ � � �� � �
� � �
äÁèÁÕ¤èÒ
�
SCMA 162 Cal. & Intro. to ODE. – p.24/30
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
Calculating Limits
Ex ¨§áÊ´§ÇèÒ � � �� � �
� �� � �
� ��
�
� �
Soln àÃÒäÁèÊÒÁÒöãªéÊٵà � � �� ��
� � �� � � �� � � � � �� ��
� �� �
� � �� ��
� ��
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
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
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
� � �
� � � �� �� � �
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
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