What is a Limit
Ms. Chathuri Ranawaka
(B.Sc.(Sp.) in Math. (Hons)(USJP−SL), M.Sc.(UoC−SL))
Sri Lanka Technology Campus
What is a limit ?
What is a limit ?
• One of the most basic and fundamental ideas of calculus
is limits.
What is a limit ?
• One of the most basic and fundamental ideas of calculus
is limits.
• Limits allow us to look at what happens in a very, very
small region around a point.
What is a limit ?
• One of the most basic and fundamental ideas of calculus
is limits.
• Limits allow us to look at what happens in a very, very
small region around a point.
• Two of the major formal definitions of calculus depend
on limits
The idea of Limit…….
Consider finding the area of a circle :
The idea of Limit…….
Consider finding the area of a circle :
n=3
n=4
n = 5 ……………………………........ n = 12
The idea of Limits…….
Consider finding the area of a circle :
n=3
n=4
n = 5 ……………………………........ n = 12
The idea of Limits…….
Let 𝒇 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐.
Fill the table.
𝒙
1.0
1.5
1.8
1.9
1.95
1.99
1.995
1.999
𝒇(𝒙)
The idea of Limits…….
Let 𝒇 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐.
Fill the table.
𝒙
𝒇(𝒙)
1.0
2.000000
1.5
2.750000
1.8
3.440000
1.9
3.710000
1.95
3.852500
1.99
3.970100
1.995
3.985025
1.999
3.997001
The idea of Limits…….
Let 𝒇 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐.
Fill the table.
𝒙
𝒇(𝒙)
𝒙
1.0
2.000000
3.0
1.5
2.750000
2.5
1.8
3.440000
2.2
1.9
3.710000
2.1
1.95
3.852500
2.05
1.99
3.970100
2.01
1.995
3.985025
2.005
1.999
3.997001
2.001
𝒇(𝒙)
The idea of Limits…….
Let 𝒇 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐.
Fill the table.
𝒙
𝒇(𝒙)
𝒙
𝒇(𝒙)
1.0
2.000000
3.0
8.000000
1.5
2.750000
2.5
5.750000
1.8
3.440000
2.2
4.640000
1.9
3.710000
2.1
4.310000
1.95
3.852500
2.05
4.152500
1.99
3.970100
2.01
4.030100
1.995
3.985025
2.005
4.015025
1.999
3.997001
2.001
4.003001
The idea of Limits…….
Let 𝒇 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐.
Fill the table.
𝒙
𝒇(𝒙)
𝒙
𝒇(𝒙)
1.0
2.000000
3.0
8.000000
1.5
2.750000
2.5
5.750000
1.8
3.440000
2.2
4.640000
1.9
3.710000
2.1
4.310000
1.95
3.852500
2.05
4.152500
1.99
3.970100
2.01
4.030100
1.995
3.985025
2.005
4.015025
1.999
3.997001
2.001
4.003001
Guess what is 𝒇 𝒙 as 𝒙 approaches 2?
The idea of Limits…….
Let 𝒇 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐.
Guess what is 𝒇 𝒙 as 𝒙 approaches 2?
“the limit of the function 𝒇 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐 as 𝒙 approaches 2 is
equal to 4.”
The idea of Limits…….
Let 𝒇 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐.
Guess what is 𝒇 𝒙 as 𝒙 approaches 2?
“the limit of the function 𝒇 𝒙 = 𝒙𝟐 − 𝒙 + 𝟐 as 𝒙 approaches 2 is
equal to 4.”
Definition of a Limit of a function
f(x)
(x,f(x))
No matter how x approaches a,
f(x) approaches L.
L
f(x)
(x,f(x))
x
a
x
x
Definition of a Limit of a function
f(x)
(x,f(x))
No matter how x approaches a,
f(x) approaches L.
L
f(x)
(x,f(x))
x
a
x
x
If as x approaches a (without actually attaining the value a), f(x)
approaches the number L, then we say that
“L is the limit of f(x) as x approaches a”,
Definition of a Limit of a function
f(x)
(x,f(x))
No matter how x approaches a,
f(x) approaches L.
L
f(x)
(x,f(x))
x
a
x
x
If as x approaches a (without actually attaining the value a), f(x)
approaches the number L, then we say that
“L is the limit of f(x) as x approaches a”,
and write
lim f ( x)  L
xa
Left & Right Hand Limits
lim f ( x)
lim f ( x)
xa
xa
2
-5
lim x 3  0
x 0
5
-2
lim x 3  0
x 0
Left & Right Hand Limits
Ex: Consider the graph of 𝑓 𝑥 given below.
Left & Right Hand Limits
Ex: Consider the graph of 𝑓 𝑥 given below.
Find left and right hand limits of
𝑓 𝑥 as 𝑥 approaches 0.
Left & Right Hand Limits
Ex: Consider the graph of 𝑓 𝑥 given below.
Find left and right hand limits of
𝑓 𝑥 as 𝑥 approaches 0.
lim 𝑓 𝑥 =?
𝑥→0−
lim 𝑓 𝑥 =?
𝑥→0+
Left & Right Hand Limits
Ex: Consider the graph of 𝑓 𝑥 given below.
Find left and right hand limits of
𝑓 𝑥 as 𝑥 approaches 0.
lim 𝑓 𝑥 = 1
𝑥→0−
lim 𝑓 𝑥 = 0
𝑥→0+
Limit of a function at a point
A function 𝒇 𝒙 has a limit as 𝒙 approaches 𝒄
Limit of a function at a point
A function 𝒇 𝒙 has a limit as 𝒙 approaches 𝒄 if and only if it has
right and left hand limits
Limit of a function at a point
A function 𝒇 𝒙 has a limit as 𝒙 approaches 𝒄 if and only if it has
right and left hand limits and these one sided limits are equal:
Limit of a function at a point
A function 𝒇 𝒙 has a limit as 𝒙 approaches 𝒄 if and only if it has
right and left hand limits and these one sided limits are equal:
Ex:
50
Does the limit exist for this
function as 𝑥 approaches 15
?
40
30
20
10
0
5
10
x
15
20
Limit of a function at a point
A function 𝒇 𝒙 has a limit as 𝒙 approaches 𝒄 if and only if it has
right and left hand limits and these one sided limits are equal:
Ex:
50
Does the limit exist for this
function as 𝑥 approaches 15
?
40
30
20
lim 𝑓 𝑥 =
𝑥→15−
10
0
5
10
x
15
20
lim 𝑓 𝑥 =
𝑥→15+
Limit of a function at a point
A function 𝒇 𝒙 has a limit as 𝒙 approaches 𝒄 if and only if it has
right and left hand limits and these one sided limits are equal:
Ex:
50
Does the limit exist for this
function as 𝑥 approaches 15
?
40
30
20
lim 𝑓 𝑥 = 20
𝑥→15−
10
0
5
10
x
15
20
lim 𝑓 𝑥 = 36
𝑥→15+
Limit of a function at a point
A function 𝒇 𝒙 has a limit as 𝒙 approaches 𝒄 if and only if it has
right and left hand limits and these one sided limits are equal:
Ex:
50
Does the limit exist for this
function as 𝑥 approaches 15
?
40
30
20
lim 𝑓 𝑥 ≠ lim+ 𝑓 𝑥
𝑥→15−
10
0
5
10
x
15
20
𝑥→15
Limit of a function at a point
A function 𝒇 𝒙 has a limit as 𝒙 approaches 𝒄 if and only if it has
right and left hand limits and these one sided limits are equal:
Ex:
50
Does the limit exist for this
function as 𝑥 approaches 15
?
40
30
20
lim 𝑓 𝑥 ≠ lim+ 𝑓 𝑥
𝑥→15−
10
0
5
10
x
15
20
𝑥→15
So, lim 𝑓 𝑥 DNE
𝑥→15
Limit of a function at a point
Ex: Find the limits of following functions as 𝑥 approaches 1
Limit of a function at a point
Ex: Find the limits of following functions as 𝑥 approaches 1
𝒇 𝒙 =
(𝒙 − 𝟏)(𝒙 + 𝟏)
(𝒙 − 𝟏)
g 𝒙 =
𝒙 + 𝟏,
𝟏,
𝒙≠𝟏
𝒙=𝟏
Limit of a function at a point
Ex: Find the limits of following functions as 𝑥 approaches 1
lim 𝒇(𝒙) = 𝟐
𝒙→𝟏
lim 𝒉(𝒙) = 𝟐
lim 𝒈(𝒙) = 𝟐
𝒙→𝟏
𝒙→𝟏
Limit of a function at a point
Ex: Find the limits of 𝑓 𝑥 as 𝑥 approaches 1, 2, 3, and 4.
Limit of a function at a point
Notice:
 Limit is a number.
 The limit can exist even when the function is not defined at a point or has a
value different from the limit.