Sequencelist of numbers

Sequence a ar as 94 with a definiteorder

Visualizations Alternate notation

I Eau or ann _I

12 as0 a a 2 a

a

Fundamental Examples 1 auction y tcxL

IN 1,2 3 3 tch

Imputantpeitiesy Eau is bounded above it there exists 14 such that

Man C M 7 u all n

4 Eau is bounded below it there exists N such that

an N 7 u all U Iau is bouncted if it is both bounded above and

below K

tK

Ly Eau is in it au Eau 7 u all n

S Eau is decreasing it an an 7 u all nI

12µsIf au is increasing decreasing we say it is monotone

We sayda is eventually bounded monotone

so

it an Nis For some N

F bounded monotone f n bounded monotone

Limitquencesau sequence an approaches L as a growsInformal positively without bound

measures how close toim an L we are

n s x Given E o there existsI ea.se N such that law Ll E

Limit tu all u N Ta number means an is in

Visualizations L E Lt En Lsu IL E 1

C E q La a L E

µan anti N

RemarksIn both cases

We have usual definitions of Limit DIVE

in an N Lim an an a u s

Lim 7cal L fol a Lim 7cal I Ic s j u a q

Limit of sequenceLimit of function

nExample an U t l an f u where

1 X1 L HospitalCt l

L 1im 7 ex Lim I Lim au I

i nx a x a

Important Theorems

If Lim an L and F is continuous at Ln s

Elen Lim 1 Can 7 Lim au 7 Ln n s

squeeze Theorem If Cerentally an E bn E caElen

Lim an Lim Cu L Lim bn Lun s a s

Monotone Convergence Theorem

au bounated and monotone Lim an L For someu

number LPicture

9 are aF.in

L M c upper bound9must be some limit point with LEM