, Limits
Notation :
lim f(x) f(a) =
x+ a
Eg
:(3)
(c + -a)
:Im
Eg =
( + --
= -
1
hm (y)- y
* the limit of a constant is that constant
Left-Hand Limits
Im f(x) =
9
x -
i
shows you come from -ve side (LIS)
Right -
Hand Limits
Im FG) =
a
+
x- 1
When
* LH tim # RH tim , the tim does not exist (jump discontinuity)
3
Eg :
Im 2x-2
eim z
=
-S
2 > 2-
him +
z
=
Co
Oblique asymptote ,
11- 2
(denom)(OA) + remander -
numerato
-
6 7 0
:. limit does not exist
Eg2 :
Limi (8) Steps :
o
O
limits
x- -
2
I factorise to simpler form
+ z)(x 2)
(x
.
-
tim · cancels sub in 1 value
#2 2 .
x - -
2
3 .
LH us RH limi
tim (x -2) = -
&
x) + -
2
Limits of so
Steps
x 6x2-
+ 3x -
1 (1 %(x + S(x 'ki) -
-
him ( (b + ki (2) i) take out LCD of dena's highest degree
=
eim 3x3 + 72 -
10 = -
1
x + x x - x
2) factorise ,
Cancel ,
etc
1- 0
=
lim 3+ 0 3) Determine
x- D
=
em
I ↳
x- d
Notation :
lim f(x) f(a) =
x+ a
Eg
:(3)
(c + -a)
:Im
Eg =
( + --
= -
1
hm (y)- y
* the limit of a constant is that constant
Left-Hand Limits
Im f(x) =
9
x -
i
shows you come from -ve side (LIS)
Right -
Hand Limits
Im FG) =
a
+
x- 1
When
* LH tim # RH tim , the tim does not exist (jump discontinuity)
3
Eg :
Im 2x-2
eim z
=
-S
2 > 2-
him +
z
=
Co
Oblique asymptote ,
11- 2
(denom)(OA) + remander -
numerato
-
6 7 0
:. limit does not exist
Eg2 :
Limi (8) Steps :
o
O
limits
x- -
2
I factorise to simpler form
+ z)(x 2)
(x
.
-
tim · cancels sub in 1 value
#2 2 .
x - -
2
3 .
LH us RH limi
tim (x -2) = -
&
x) + -
2
Limits of so
Steps
x 6x2-
+ 3x -
1 (1 %(x + S(x 'ki) -
-
him ( (b + ki (2) i) take out LCD of dena's highest degree
=
eim 3x3 + 72 -
10 = -
1
x + x x - x
2) factorise ,
Cancel ,
etc
1- 0
=
lim 3+ 0 3) Determine
x- D
=
em
I ↳
x- d
