Limits Theory Notes

f Cx )

Limits


i
-




Pca
q
)




at
~
L
L - - - ,




,h
I
-




a
f C x ) =
L I I K


p p →
" t RH

If the function is continuous
←
at a


tea f Gc ) =
fca ) =L




0ne-sidedlim

RH Limit LH Limit


¥ at
fcx ) =L tea - f Gc ) =L


x > a xLa




Relationship between limit and one - sided limit


×h→a+fCx ) =L
,¥a+ fad ¥a Stx )
}¥afCx
F
) =L
.




AND tea -
Sec ) =L THEN tea FCK ) DNE




#c→:
5- Cx )
Limit at
-
infinity
- -
- . -
- - -
-
in - - - -
- -




⇐ ±
f Cx ) =L
as

X




f- GD
' ⇒ → A
Infinite Limit ^ i

#
I




tea fcx ) = IN
I
I 2x
a

,
I

I




Vertical Asymptote

, Limit Laws
-




If TEA fed & xthagcx) both exist THEN


¥24 ] 2¥
tea I
I=qfEafc⇒
①
=
a
" =
4
e.g .




constant




# gcx ) ¥+242 txhze
"


② tea ffcsdtgcx ) ] = she,
a
fad ±
a
tea [ site ] =



e.g .




'

= 4 x e




schfaffcsd
god ] ¥afCx) # agcx ) ¥2 [24×2]=1%2 If
'
"

③ .
= x
e.g .
x zx

=
4×4=16




④
¥aKd=÷÷F ⇒
e
.
.÷s÷=÷÷÷s=÷=¥ -

Xo

"



)
"




] I Ea ¥32 )
"


⑤ Ihsa I Stx = HH
e.g
.
¥2 x
"
=
"




)
"




( ¥52 ( )
¥2
"

[22+2×+3] = x 2+2×+3



Isaiah
"

⑥ Fsa xD
#
"


[ "I
-
=




-

-

East