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Lecture

# Limits Cheat Sheet

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School
University of Ottawa
Department
Mathematics
Course
MAT1320
Professor
Kirill Zaynullin
Semester
Fall

Description
Calculus Cheat Sheet Limits Definitions Precise Definition : We say lixﬁa x( )L if Limit at Infinity : We say xﬁ¥ f ( ) L if we for every e > 0 there is ad > 0such that can make f x( ) close to L as we want by whenever 0 < x-a a. There is a similar definition for lim f x = -¥ xﬁa ( ) Left hand limit : lim - x( )L. This has the xﬁa except we make f ( )rbitrarily large and same definition as the limit except it requires negative. x < a. Relationship between the limit and one-sided limits lim f ( ) L Þ lim f x + l( )f x = - ( ) lim+f ( ) lim f-x =( )Þ lim f x = L ( ) xﬁ a xﬁa xﬁa xﬁa xﬁa xﬁa lim f ( ) lim f x ( )im f x Does( )t Exist xﬁa+ xﬁa- xﬁa Properties Assume lim f( ) and limg ( ) both exist and c is any number then, xﬁa xﬁa 1. limØcf ( ) = clim f x( ) Ø f x ø lim f ( ) xﬁa º ß xﬁa 4. lim Œ ( ) œ= ﬁ a provided lim g( ) „ 0 ﬁ a g( ) limg x( ) xﬁa º ß ﬁ a 2. limØ º x( )g x ( )ßlim f x ( )mg x ( ) n n xﬁa xﬁa xﬁa 5. lixﬁaº x( )ß limºxﬁa ( )ß 3. limØ f x g x ø = lim f x limg x 6. lim Øn f ( )ø = lim f x( ) xﬁa º ( ) ( ) ß xﬁa ( ) xﬁa ( ) xﬁaº ß xﬁa Basic Limit Evaluations at – ¥ Note : sgn a =1 if a > 0 and sgn a = -1 if a < 0 . ( ) ( ) 1. lim e = ¥ & lim e = 0 5. n even : lim x = ¥ xﬁ¥ xﬁ- ¥ xﬁ–¥ n n 2. lxﬁ¥ln ( ) ¥ & xﬁ0m+ln( )= -¥ 6. n odd : lxﬁ¥x = ¥ & limxﬁ- ¥-¥ n 3. If r > 0then lim b = 0 7. n even : lim a x +L+bx+c = sgn a ¥ ( ) xﬁ¥ x r xﬁ–¥ r 8. n odd : lim ax +L+bx+c = sgn a ¥ ( ) 4. If r > 0 and x is real for negative x xﬁ¥ b 9. n odd : lim ax +L+cx+d = -sgn a ¥ ( ) thenxﬁ-¥ r = 0
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