Lecture #: 03: "Limits of Functions" with in a course on Applied Calculus offered at Faculty of Engineering, University of Central Punjab By: Prof. Muhammad Rafiq.

1. Applied Calculus
1st
Semester
Lecture -03
Muhammad Rafiq
Assistant Professor
University of Central Punjab
Lahore Pakistan

2. LIMIT OF A FUNCTION
1) Definition
A fuction is said to have a limit if is defined for
larger value of and as , .
i.e
∞
EXAMPLES:
IF
Then

3. Collectively,
∞
2) DEFINITION:
A function is said to have a limit if is defined for
all values of close to and as ,

5. RULES FOR FINDING LIMITS:
THEOREM 1 (PROPERTIES OF LIMITS):
1) , where c is a constant.
2)
3)
4)
5)
6) →
→
,provided
7) If m,n are integers then
provided
is real.

6. THEOREM 2:
1). If p(X) + +……+
Then + +…….
THEOREM 3:
If P (x) and Q(x) are polynomials and ,then
( )
( )

7. THEOREM 4: (SANDWITCH THEOREM)
Suppose that g(x) for all x in an open interval
containing ‘c’ except possibly at x itself .suppose also that
, then .
EXAMPLES:
1)
1
So by sandwitch theorem

8. 2)
=
=
= 1+1= 2
3)
=
=
=

9. =
= 3
=
=
=
=
=

10. 5)
= -
= -
= -
= -
= - ∞
6)
=

11. =
= -2+3
= 1
=
= -
= - (-2+3)
= -1
One Sided Limit

12. x→ ; f(x) → 4 (Left hand limit)
x→ ; f(x) → 4 (Right hand limit)
So as,

13. Theorem: (One sided vs. two sided limits)
A function f(x) has a limit as ‘x’ approaches to ‘c’ if and only if
it has left hand limit and right hand limit at x=c and both are
equal.
i.e
Example Given that

14. So by Sandwich Therem
Example Q(37) Ex 1.2
f(x)= at x=1
f(x+h)=

15. But we have to find answer at x=1
So,
Example

16. Since,
Example
f(x) = |x-5|, find
f(x)=|x-5|=

17. Since,
Example
Find

18. Example
∞
∞

19. Example
∞
∞

20. Example
∞
∞
does not exist
Exercise 1.2
Q (1-44) Odd Questions
Exercise 1.4
Q (1-46) Odd Questions