-
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
"
'
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
-
wise
*i÷÷i÷o÷÷÷±÷÷÷
definitions *
*
only replace abs Val wipe ice wise
-
,
. .
→ x'it iii.iii.eat:*:
'
y . .
*
Ye:*:
"
x'Ez
'
) '
"
lying
"
limit DNE
-
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
"
""" "'
-
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¥
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
"
'
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
-
wise
*i÷÷i÷o÷÷÷±÷÷÷
definitions *
*
only replace abs Val wipe ice wise
-
,
. .
→ x'it iii.iii.eat:*:
'
y . .
*
Ye:*:
"
x'Ez
'
) '
"
lying
"
limit DNE
-
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
"
""" "'
-
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¥
