Bangladesh Army University of Science and Technology (BAUST)

Department of Arts and Sciences

Lecture Topic: Limit and Continuity

Function: A function 𝑓 is a rule that associates a unique output with each input. If the input is
denoted by 𝑥, then the output is denoted by 𝑓(𝑥) (read 𝑓 of 𝑥).

Concept of function and its domain and range

Limit: If the value of 𝑓(𝑥) become arbitrarily close to a single number 𝑙 as the values of a
variable 𝑥 approaches to a from both side of 𝑎 then 𝑙 is called the limit of the function of 𝑓(𝑥). It
is denoted by lim 𝑓(𝑥) = 𝑙.
𝑥→𝑎

Behavior of a function near a point.

𝒙𝟐 −𝟏
How does the function 𝒇(𝒙) = behave near 𝒙 = 𝟏?
𝒙−𝟏
Solution: The given function is defined for all real numbers 𝑥 except 𝑥 = 1 (we cannot divide by
zero). For any 𝑥 ≠ 1, we can simplify the formula by factoring the numerator and canceling
common factors.

Irfan Raju
Assistant Professor (Mathematics)
Department of Arts and Sciences, BAUST
 𝑥 2 − 1 (𝑥 + 1)(𝑥 − 1)
𝑓(𝑥) = = =𝑥+1
𝑥−1 (𝑥 − 1)
The graph of 𝑓(𝑥) is thus the line 𝑦 = 𝑥 + 1 with the
point (1,2) removed. This removed point is shown as a
“hole” in the figure 2. Even though 𝑓(1) is not defined, it
is clear that we can make the value of 𝑓(𝑥) as close as we
want to 2 by choosing 𝑥 close enough to 1 (see the
following table-1)

Table-1:
𝒙 approaches to 1 from left Values of 𝒇(𝒙) 𝒙 approaches to 1 from Values of 𝒇(𝒙)
left
0.9 1.9 1.1 2.1
0.99 1.99 1.01 2.01
0.999 1.999 1.001 2.001
0.999999 1.999999 1.000001 2.000001
↓ ↓ ↓ ↓
1 2 1 2

We say that 𝑓(𝑥) approaches limit 2 as 𝑥 approaches, and write

𝑥 2 −1
lim 𝑓(𝑥) = 2 or lim =2
𝑥→1 𝑥→1 𝑥−1

Continuity: A function f is said to be continuous at 𝑥 = 𝑎 provided the following conditions are

satisfied:

i. 𝑓(𝑎) is defined
ii. lim 𝑓(𝑥) exists
𝑥→𝑎
iii. lim 𝑓(𝑥) = 𝑓(𝑎)
𝑥→𝑎

or

Continuity: A function 𝑓(𝑥) said to be continuous at 𝑥0 if for given ℇ > 0 there exist 𝛿 > 0
such that

|𝑓(𝑥) − 𝑓(𝑥0 )| < ℇ Whenever |𝑥 − 𝑥0 | < 𝛿

Irfan Raju
Assistant Professor (Mathematics)
Department of Arts and Sciences, BAUST
Symbolically,

lim 𝑓(𝑥) = 𝑓(𝑥0 ).

𝑥→𝑥0

Example:

𝑥2 , 𝑥<1
Q. A function 𝑓(𝑥) is defined as 𝑓(𝑥) = { 2.5 , 𝑥 = 1 . Does lim 𝑓(𝑥) exist?
𝑥→1
𝑥2 + 2 , 𝑥>1

Solution: Left hand limit= 𝑙𝑖𝑚− 𝑓(𝑥) = 𝑙𝑖𝑚− 𝑥 2 = 1

𝑥→1 𝑥→1

Right hand limit= 𝑙𝑖𝑚+ 𝑓(𝑥) = 𝑙𝑖𝑚+(𝑥 2 + 2) = 3

𝑥→1 𝑥→1

Since, Left hand limit ≠ Right hand limit

So lim 𝑓(𝑥) does not exist.

𝑥→1

Irfan Raju
Assistant Professor (Mathematics)
Department of Arts and Sciences, BAUST
 3
3 + 2𝑥 , −2≤𝑥 <0
3
Q. Test the continuity of the function 𝑓(𝑥) = 3 − 2𝑥 , 0≤𝑥<2 at the point
3
−3 − 2𝑥 , 𝑥≥2
{
3
𝑥 = 0 and 𝑥 = 2.

Solution: When 𝑥 = 0, 𝑓(𝑥) = 3 − 2𝑥

∴ 𝑓(0) = 3 − 2.0 = 3

Left hand limit= 𝑙𝑖𝑚− 𝑓(𝑥) = 𝑙𝑖𝑚− (3 + 2𝑥) = 3

𝑥→0 𝑥→0

Right hand limit= 𝑙𝑖𝑚+ 𝑓(𝑥) = 𝑙𝑖𝑚+( 3 − 2𝑥) = 3

𝑥→0 𝑥→0

Since, 𝑓(0) = 𝑙𝑖𝑚+ 𝑓(𝑥) = 𝑙𝑖𝑚− 𝑓(𝑥)

𝑥→0 𝑥→0

3
So 𝑓(𝑥) is continuous at 𝑥 = 0, Similarly for 𝑥 = 2.

𝑥, 𝑥>0
Q. A function 𝑓(𝑥) is defined as 𝑓(𝑥) = { 0 , 𝑥 = 0 . Does lim 𝑓(𝑥) exist?
𝑥→0
−𝑥 , 𝑥<0
1
1 + 2𝑥 , −2≤𝑥 <0
1
Q. A function 𝑓(𝑥) is defined as 𝑓(𝑥) = 1 − 2𝑥 , 0≤𝑥<2 . Does lim1 𝑓(𝑥)
𝑥→
1 2
−1 + 2𝑥 , 𝑥≥2
{
and lim 𝑓(𝑥) exist?
𝑥→0

−𝑥 2 , 𝑥≤0
5𝑥 − 4, 0<𝑥≤1
Q. Test the continuity of the function 𝑓(𝑥) = { 2 at 𝑥 = 0, 𝑥 = 1 and
4𝑥 − 3𝑥, 1<𝑥<2
3𝑥 + 4, 𝑥≥2
𝑥 = 2.

𝑥+1, 𝑥<2
Q. Test the continuity of the function 𝑓(𝑥) = { 𝑥2 , 𝑥=2 at the point = 2 .
2𝑥 − 1 , 𝑥>2
𝑥+1, 𝑥<2
Q. Test the continuity of the function 𝑓(𝑥) = { at the point = 2 .
2𝑥 − 1 , 𝑥≥2

Q. Show that the function 𝑓(𝑥) = |𝑥| + |𝑥 − 1| + |𝑥 − 2| is continuous at the points 𝑥 = 0, 1, 2.

Irfan Raju
Assistant Professor (Mathematics)
Department of Arts and Sciences, BAUST