1. ENGINEERING MATHEMATICS I
(BWM 10103)
Seksyen 2

2. PERSONAL DETAILS Name: Dr. Fariza Mohamad Room: Block C15, Room 17

3. LIMITS AND CONTINUITY

4. CONTENTS Definition of limits Evaluation of limits Right sided limit
Left sided limit
Two-sided limit
3. Infinite limit
4. Technique of evaluating limits
a) Limit at a point
b) Limit at infinity .

5. CONCEPT OF LIMITS The limit of a function refers to behavior of a
function as the independent variable approaches
a given value.
If the limit of function f (x) as x approaches the point a is
the value L, then it is denoted as
In simpler terms, the definition says that as x get closer
and closer to a then f (x) must getting closer and
closer to L.
* Limit can be establish from tables and graphs. .

6. CONCEPT OF LIMITS Using tables
Estimate the value of the following limit.
1. Choose values of x that get closer to closer to 2
2. substitute the values into Doing this, gives
the following table: x
1.5
1.9
1.999
1.9999 2
2.0001
2.001
2.01
2.5
f (x)
5.0
4.158
4.002
4.0002
3.9999
3.999
3.985
3.4
Approach to 2 from the left side
Approach to 2 from the right side

7. CONCEPT OF LIMITS The function is going to 4 as x approaches 2, then
Important !!
Using tables of values to guess the value of limits is simply
not a good way to get the value of a limit but they
can help us get a better understanding of what limits are.

8. CONCEPT OF LIMITS One-sided limits: Right side
Only look for limit at right side of the point.
Right-handed
x approaches a from
right hand side.
We write as

9. CONCEPT OF LIMITS One-sided limits : Left side
Only look for limit at left side of the point.
Left-handed
x approaches a from
left hand side.
We write as

10. CONCEPT OF LIMITS Two-sided limits The limit
is called two-sided limit.
>> Requires value of limits from right and left sides.
Two-sided limit of f (x) only exist if
Then,
Example: One and Two-sided limits

11. CONCEPT OF LIMITS
Example: One and Two-sided limits

12. INFINITE LIMITS Infinite limits Infinite limits occur when and .
All limits that have infinite limit = limit does not exist.
Example: Infinite limits

13. TECHNIQUES OF COMPUTING LIMITS
limits at a point
Consider first limits at a point which is
“limits of f (x) when x approaches any point a”.
Some basic properties of limits
Let a and k is any real number

14. TECHNIQUES OF COMPUTING LIMITS
limits at a point

15. TECHNIQUES OF COMPUTING LIMITS
limits at a point
We will discuss algebraic techniques for computing limits of:-
Polynomial functions
Rational functions
Radical functions, (function involve )
Piecewise functions

16. TECHNIQUES OF COMPUTING LIMITS
limits at a point
Polynomial functions
Let f (x) is polynomial function
Then,
Example: Polynomial function

17. TECHNIQUES OF COMPUTING LIMITS
limits at a point
Polynomial example
Evaluate the following: a) b)

18. TECHNIQUES OF COMPUTING LIMITS
limits at a point
Rational functions
Let f (x) is rational function
, with d(x) and n(x) polynomial function
Then,
,provided that
But how if ???
does not exist. 1 If 2 If
, we have to factor d(x) and n(x)
(by common factor (x – a) ) then cancel out
(SIMPLIFIED THE FUNCTION).

19. TECHNIQUES OF COMPUTING LIMITS
limits at a point
Rational functions
Evaluate the following:

20. TECHNIQUES OF COMPUTING LIMITS
limits at a point
Radical functions, (function involve )
Let f (x) is radical function
, with e(x) and m(x) radical function
Then,
For
Rationalize is:-
Rationalize???
,provided that
But how if ???
does not exist. 1 If 2 If
, rationalize
e (x) or m (x)

21. TECHNIQUES OF COMPUTING LIMITS
limits at a point
Radical functions, (function involve )
Example: Radical function

22. TECHNIQUES OF COMPUTING LIMITS
limits at a point
Piecewise functions
Piecewise function is defined by
The limits of piecewise function is obtained by finding the
one-sided limits first.

23. TECHNIQUES OF COMPUTING LIMITS
limits at a point
Piecewise functions
Example: Piecewise function

24. 4. Technique of evaluating limits a) Limit at a point
TECHNIQUES OF COMPUTING LIMITS
limits at a point
Summary:
Definition of limits
Evaluation of limits
Right sided limit
Left sided limit
Double sided limit
3. Infinite limit
4. Technique of evaluating limits
a) Limit at a point

25. Lecture

26. TECHNIQUES OF COMPUTING LIMITS
limits at infinity
Limits of f (x) when x increase without bound ( )
or x decrease without bound ( ).
Some basic properties of limits
Let k is any real number

27. TECHNIQUES OF COMPUTING LIMITS
limits at infinity

28. TECHNIQUES OF COMPUTING LIMITS
limits at infinity
We will discuss algebraic techniques for computing limits of:-
Polynomial functions
Rational functions
Radical functions, (function involve )

29. TECHNIQUES OF COMPUTING LIMITS
limits at infinity
Polynomial functions
Let f (x) is polynomial function
Then,
depends on the highest degree term and
consider

30. TECHNIQUES OF COMPUTING LIMITS
limits at infinity
Polynomial functions
Example: Polynomial function

31. TECHNIQUES OF COMPUTING LIMITS
limits at infinity
Rational functions
Let f (x) is rational function
Then,
Step 1
Divide each term in n(x) and d(x) with
the highest power of x.
Step 2
Substitute to find the answer

32. TECHNIQUES OF COMPUTING LIMITS
limits at infinity
Rational functions
Example: Rational function

33. TECHNIQUES OF COMPUTING LIMITS
limits at infinity
Radical functions, (function involve )
Let f (x) is radical function
Try substitute the value of into the function. If can’t
find the answer, consider these cases:
Case 1
Case 2
f (x) = m (x) only
Rationalize m (x)
Step 1
Divide m(x) and e(x) with the highest
power of x
Step 2
Substitute to find the answer
Special case?
Use concept

34. TECHNIQUES OF COMPUTING LIMITS
limits at infinity
Radical functions, (function involve )
Example: Radical function
(d)

35. CONTINUITY We perceive the path of moving object as an unbroken
curve without gaps, breaks or holes. In this section,
we translate the unbroken curve into a precise
mathematical formulation called continuity.
A function f (x) is said to be continuous at x = a if
Provided that EXIST and DEFINED.