The document defines the limit of a function. It states that a function f(x) is defined on a bounded domain R and the limit is evaluated at a point x0 as x approaches x0 from the positive side. It introduces ε and δ such that if |f(x) - f(x0)| < ε when 0 < |x - x0| < δ, then the limit of f(x) as x approaches x0 is f(x0). It provides an example of the limit of f(x) = 1/x as x approaches 0 and explains that the limit does not exist since ε is undefined.

1. Definition of Limit
Arun Umrao
October 13, 2021
https://sites.google.com/view/arunumrao
Arun Umrao Definition of Limit

2. Definition of Limit
Before finding about limit of a function, following assumptions are
made about the function.
f (x) is function of x.
Arun Umrao Definition of Limit

3. Definition of Limit
Before finding about limit of a function, following assumptions are
made about the function.
f (x) is function of x.
f (x) is defined in bounded and closed domain R.
Arun Umrao Definition of Limit

4. Definition of Limit
Before finding about limit of a function, following assumptions are
made about the function.
f (x) is function of x.
f (x) is defined in bounded and closed domain R.
Following conditions are assumed for the function whose limit is
being obtained at a point.
Point at which limit is to be found is x0.
Arun Umrao Definition of Limit

5. Definition of Limit
Before finding about limit of a function, following assumptions are
made about the function.
f (x) is function of x.
f (x) is defined in bounded and closed domain R.
Following conditions are assumed for the function whose limit is
being obtained at a point.
Point at which limit is to be found is x0.
Approaching to point x0 from x from positive side (let).
Arun Umrao Definition of Limit

6. Definition of Limit
Before finding about limit of a function, following assumptions are
made about the function.
f (x) is function of x.
f (x) is defined in bounded and closed domain R.
Following conditions are assumed for the function whose limit is
being obtained at a point.
Point at which limit is to be found is x0.
Approaching to point x0 from x from positive side (let).
Function values are f (x0) and f (x) respectively.
Arun Umrao Definition of Limit

7. Definition of Limit
Before finding about limit of a function, following assumptions are
made about the function.
f (x) is function of x.
f (x) is defined in bounded and closed domain R.
Following conditions are assumed for the function whose limit is
being obtained at a point.
Point at which limit is to be found is x0.
Approaching to point x0 from x from positive side (let).
Function values are f (x0) and f (x) respectively.
Here
x and x0 are neighbouring points
Arun Umrao Definition of Limit

8. Definition of Limit
Choose two definite, small and real arbitrary numbers ǫ > 0 and
δ > 0. These two numbers should satisfy the following conditions.
|f (x) − f (x0)| < ǫ
Arun Umrao Definition of Limit

9. Definition of Limit
Choose two definite, small and real arbitrary numbers ǫ > 0 and
δ > 0. These two numbers should satisfy the following conditions.
|f (x) − f (x0)| < ǫ
0 < |x − x0| < δ
Arun Umrao Definition of Limit

10. Definition of Limit
Choose two definite, small and real arbitrary numbers ǫ > 0 and
δ > 0. These two numbers should satisfy the following conditions.
|f (x) − f (x0)| < ǫ
0 < |x − x0| < δ
If this is true, then we can say that
lim
x→x0
f (x) = f (x0)
Arun Umrao Definition of Limit

11. Definition of Limit
Meaning of ǫ and δ
Data Table for function f (x) = 1/x in x ∈ [−0.4, +0.4] is given
below.
Data Table
x f (x)
-0.4 -2.5
-0.3 -3.3
-0.2 -5.0
-0.1 -10.0
0.0 Div by zero
0.1 10.0
0.2 5.0
0.3 3.3
0.4 2.5
Arun Umrao Definition of Limit

12. Definition of Limit
Meaning of ǫ and δ
In following figure, function f (x) = 1/x is plotted in x ∈ [−2, +2].
Data Plot
10
20
−10
−20
1
−1
−2
x
f (x)
Arun Umrao Definition of Limit

13. Definition of Limit
Meaning of ǫ and δ
In following figure, function f (x) = 1/x is plotted in x ∈ [−2, +2].
Data Plot
10
20
−10
−20
1
−1
−2
x
f (x)
Data plot
10
20
−10
−20
−1
x
f (x)
|ǫ|
x
|δ|
x0
Arun Umrao Definition of Limit

14. Definition of Limit
Meaning of ǫ and δ
Meaning of delta
Let the arbitrary point from right hand side is x while it is
approaching to x0 = 0. Choose x = 0.01, now absolute difference
(difference is difference either positive or negative) between two
consecutive neighbouring points is
δ = |0.01 − 0| = 0.01
Here, δ is small and definite hence it is acceptable value.
Arun Umrao Definition of Limit

15. Definition of Limit
Meaning of ǫ and δ
Meaning of ǫ
Function values at points are at x = 0.1 and x0 = 0 are
f (0.01) =
1
0.01
= 100; f (0) =
1
0
= ∞
Now, difference between these two function values is
ǫ = |f (0.01) − f (0)| = ∞
Here, ǫ is undefined, it means when we move from right hand side
to x = 0 we will never get a definite function value at x = 0, so
limit at x = 0 does not exist for the given function f (x) = 1/x.
Arun Umrao Definition of Limit

16. Definition of Limit
Meaning of ǫ and δ
Meaning of ǫ and δ
It is clear that ǫ represents to the absolute difference between two
neighbouring functions and δ represents to the absolute difference
between two corresponding neighbouring points.
Arun Umrao Definition of Limit

17. Definition of Limit
Thank You
Arun Umrao Definition of Limit