MTH 253 Calculus (Other Topics)Formal Definitions …Slide 3Slide 4Intuitive Definitions …Rule 1 – Plug it In!Rule 2 – Algebraic Equivalences!Rule 3 – QuotientsSlide 9Slide 10Slide 11Rule 4 – Other Indeterminate FormsSlide 13Rule 5 – Limits of Rational Functions as xRule 6 – Transcendental FunctionsSlide 16Slide 17Slide 18MTH 253Calculus (Other Topics)Chapter 10 – Infinite SeriesSection 10.0 – Review of Methods for Evaluating LimitsCopyright © 2008 by Ron Wallace, all rights reserved.Formal Definitions …lim ( )x af x L�=Means that for all  > 0 there exists a  > 0 such that |f(x) – L| <  whenever 0<|x – a| < .1 of 3f(x)L+L-f(x)xa- a+LaFormal Definitions …lim ( )xf x L��=Means that for all  > 0 there exists an N > 0 such that |f(x) – L| <  whenever x > N.3 of 3f(x)L+L-f(x)xNLFormal Definitions …lim ( )x af x�=�Means that for all M > 0 there exists a  > 0 such that f(x) > M whenever 0<|x – a| < .2 of 3f(x)f(x)xa- a+MaIntuitive Definitions …lim ( ) 0x af x�=Means that |f(x)| is really small when x is close to a.Is it +0 or –0?lim ( )x af x�=�Means that |f(x)| is really big when x is close to a.Is it + or –?lim ( )x af x L�=Means that f(x) is really close to L when x is close to a.Is it a little more or a little less?lim ( )x af x DNE�=Means that either the limit is +, –, or it bounces around as x gets close to a.Rule 1 – Plug it In!Calculate f(a). If there are no problems, f(a) is the limit.Possible problems?Division by zero.Even roots of negatives.Evaluating a function at a value not in its domain.Indeterminate forms0/0, /, 0∙ , - 1, 00, 0lim ( )x af x�Rule 2 – Algebraic Equivalences!Simplify algebraicallyEspecially rational expressionsAdd/subtract rational expressionsLong division w/ rational expressionsRationalize numerator/denominatorlim ( )x af x�Rule 3 – Quotients( )lim( )x af xg x�Case 1:f(x)m and g(x)n as xa & m0, n 0nmxgxfax)()(limCase 2:f(x)0 and g(x) n as xa & n00)()(lim xgxfaxm over n0 over nRule 3 – Quotients( )lim( )x af xg x�Case 4:f(x) and g(x)n as xa & n0Case 3:f(x)m and g(x)0 as xa & m0DNEor or )()(lim xgxfaxDNEor or )()(lim xgxfaxm over 0 over nRule 3 – Quotients( )lim( )x af xg x�Case 6:f(x) and g(x)0 as xaCase 5:f(x)m and g(x) as xa & m00)()(lim xgxfaxDNEor or )()(lim xgxfaxm over  over 0Rule 3 – Quotients( )lim( )x af xg x�Case 7:f(x)0 and g(x) as xa0)()(lim xgxfaxCase 8: f(x)0 and g(x) 0 as xa … or …f(x) and g(x) as xa( ) '( )lim lim( ) '( )x a x af x f xg x g x� �=L’Hôpital’sRule0 over 0 over 0or over Rule 4 – Other Indeterminate FormsCase 1:f(x)0 and g(x) as xa1( )( )lim ( ) ( ) limx a x ag xf xf x g x� �=gCase 2:f(x) and g(x) as xalim ( ) ( ) lim ( )x a x af x g x h x� �- =Then useL’Hôpital’s Rulei.e. Change Algebraically0 times  minus Rule 4 – Other Indeterminate FormsCase 3: f(x)1 and g(x)  as xa … or … f(x)0 and g(x)0 as xa … or … f(x) and g(x)0 as xa1( )ln( ( )) lim lim ( )ln( ( ))( )lim ( )x ag xx af xg x f xg xx af x e e���= =Then useL’Hôpital’s Rule1 or 00 or 0Rule 5 – Limits of Rational Functions as xCase 1:m < n lim 0 mnxaxbx���+=+ggggggCase 2:m > n lim mnxaxbx���+=��+ggggggCase 3:m = n lim mnxax abx b���+=+ggggggRule 6 – Transcendental Functions2lim tanx kxpp-� �� �� �� �=+�2lim tanx kxpp+� �� �� �� �=- �( )lim cotx kxp p-� �=- �( )lim cotx kxp p+� �=+�1 of 4Rule 6 – Transcendental Functions22lim secx kxpp-� �� �� �� �=+�22lim secx kxpp+� �� �� �� �=- �2lim secx kxpp-� �� - �� �� �=- �2lim secx kxpp+� �� - �� �� �=+�2 of 4Rule 6 – Transcendental Functions( )0 2lim cscx kxp-� �=- �( )0 2lim cscx kxp+� �=+�( )2lim cscx kxp p-� �=+�( )2lim cscx kxp p+� �=- �3 of 4Rule 6 – Transcendental Functionslim 0xxe�- �= limxxe��=+�0lim 1xxe�=0lim lnxx+�=- �lim lnxx��=+�1lim ln 0xx�=4 of