Limit rules (Notes) MATH 184 (UBC)

-


HOW TO FIND ANY LIMIT AT A FINITE VALUE -




-




LIMITS -




1) Plug (substitution ) →
If # that's
in you get a
, !÷!¥ ! innit



§ !Ms
.



X't 2X -

4 =
(5)2+2 ( s) -
4 = 25 *
10-4=310 ←
innit


,




→ If
you get
Tb or
-8

¥7 → =
8- = undefined



2) factor ¥72 (¥1)
H¥¥
''
=
4 t link = x'Fa Cx -123--2+2=40



3) Get Common denominator ¥7 , i¥ ÷¥ + him ¥: = =

,




4) Open Up Paran theses (expand then Simplify) tiny limo "" limo =
=li¥o*¥ limo # 41--0+4=40

congii
"
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5) Multiply by rationalize when given rxlfi.rs#TTYFI3=x'Ii
'
'




x'Ii
'
-
-




=¥i¥ =s¥=⑤
6) Something of the form SII

¥iosm x'Io
' "
-

-

hi =D
to sine I

¥¥¥m÷÷
-
-




,
timo ¥-1
him ¥-0
.




7) Absolute Value

¥72 Y ①
redwenjiten nice
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wise




*i÷÷i÷o÷÷÷±÷÷÷
definitions *
*
only replace abs Val wipe ice wise
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,
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→ x'it iii.iii.eat:*:
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y . .




*
Ye:*:
"
x'Ez
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) '
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lying
"

limit DNE
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L .

.
.

,-




LIMITS -




example




.net#I - 4/i mlx-yxIT=tixFzcx+os
① Direct substitution notice as we
get exam
② factor
,




]
closer and closer to 2
-8
,
pugs → = - undefined
the limit approaches
" ¥72 (¥4)=4
t.im
I;÷:
"
=÷i= "


4¥
.




p;gpa
" " " to ' " 2
'
"
#
YEH
.




-1=2+2--40
examp1e#⑦ exampleHQF.F.7.ie#aYtoni*exampleHQ example#@ WE;EiII÷÷%¥÷
*




LIM X't 2×-4 Iim X3_27_ Iim t um r× -
z + zthe
expression that hast


X -13 × 3
X -13 3
Exes)
-




X -15 X -

X -19
( digests, *,

'
(5) +215) idiots :tA Kia bicaeiabtb,
tf!,Y =L!.mg#s
'

= -
4 =
-
- =
=
,
X 3
f
-




25*10-4
¥73 =r÷=¥=⑤
'

f3j=×z+s×+a x'Is
= -
=


=
init
←

=¥y
= =


,
3. +
ya ,+g=③

example# example#@ example#Q
11M ¥ -

*
for some limits , need to




!¥yo¥=+* HI ¥
"'
'
Y:p: rim:T*
*4
Fy a
:*. =L ,




:o)
, ,

¥
i:÷÷÷÷ :÷÷÷÷÷÷

}¥÷
± :+÷. o
.
.




11M -
2- Tx Gtr ) YIo=D as

foot)
.n=¥s
-




¥3,2
.
-
too


-

-


10000
"
""" "'
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n' i'
" "'

Hot
¥÷÷÷÷÷÷÷÷?
"
"⇐
-
- .


x-iywxx.cz#--ItfFxIhTz*rx)
"

no .n=o÷= -
to


4¥)=
Ilm l
-

-




x-iy-rxz.TT) #
= =

fco .oD= = -100


TRIG "

wie JW
examp1e#Q example#gt¥example#
-




""u

remember
example#
¥ho IT tan× =-3
'
""'
If sm "m



=⇐
'
5×+3
=

xd
'




=
'"
""
f- C- on) = = -
l X -10


Is:
'




¥7:
.
'

s¥÷÷÷÷÷÷÷:nmcs=e X to
-
4x 4
÷
lim f-Cx)
%f÷÷¥ :*
"m "
" To evaluate from attsrightsiou
Xy

¥no⇐gn×
,
.
;÷ . . .


=
lines ¥ -5
lim fcx)
.



.



too -3 5×+3=54) -13=80

.*=⑤I¥÷i÷÷÷÷÷:
o .




exa iii.
tim
8×+21=9×5--4
← 29's
one .
become
..
Insightat YI :* ,
, " " "



e×a3 2×-8
x to
fly) 3×-2=314) -2=100
-

-
-




Iim o
' ' ooo =


,=% lime
35+4=130
'

x IN X
-


X -14 = (
top ' Xyz
-


on bottom
degree
EXamp1e#
same
11=100-1 ÷ = 0.01
←

1000-1*0=0.001 9-3lf.TO#Xt8-FICYxD5tiIo-x-3+2tiIaxz
Ilm 9×2 3×cy×y



# 95.3¥
-



x -




Is =-D
lim 1¥ .co ,
--




X -18 I
-
3 -1¥