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# Limits Cheat Sheet

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University of Ottawa

Mathematics

MAT1320

Kirill Zaynullin

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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