1. Consider the function Note that for 1 from the right from the left
LIMIT OF A FUNCTION
Consider the function
Note that for
Investigate the behavior of f(x) as x goes very very close to 1.
from the left 1
from the right
Take x close to 1 but less than 1
Take x close to 1 but greater than 1

2. Let by assigning values of x very very close to 1 but less than 1.
0.5
0.75
0.9
0.99
0.999
0.9999
f(x) 5 6
6.5
6.8
6.98
6.998
6.9998
In symbol, we write this as
ONE-SIDED LIMIT : LEFT HAND SIDE LIMIT

3. Let by assigning values of x very very close to 1 but greater than 1.2
1.5
1.25
1.1
1.01
1.001
1.0001
f(x) 9 8
7.5
7.2
7.02
7.002
7.0002
In symbol, we write this as
ONE-SIDED LIMIT : RIGHT HAND SIDE LIMIT

4. If one-sided limits both exist and are equal
TWO-SIDED LIMIT
If one-sided limits both exist and are equal
to one another then two-sided limit exists
and is equal to the same value.
Note that function value at x = 1 does not
exists but limit of the function as x approaches
close 1 exists and is equal to 7.
So limit of a function is that number to which
the function value is going very near to as we
take x very close to 1.

5. We express this by saying that the limit of f(x) 7
x approaches 1 from the left
x approaches 1 from the right 6 8
Observe from both tables that as x gets closer to 1 (on either side of 1), f(x) gets closer to 7.
We express this by saying that the limit of f(x)
as x approaches 1 is equal to 7.
We write this in symbol as
This leads to the intuitive definition of limit.

6. INTUITIVE DEFINITION OF LIMIT
Let f(x) be defined on an open interval containing a,
except possibly at a itself. We say that the limit of
the function f(x) as x approaches a is equal to L ,
written as
if f(x) gets closer and closer to L, whenever
x gets closer and closer to a (on either side of a).
Going back to Tables 1 and 2, we see that the closer x is to 1, the closer f(x) is to 7. Also, we can make the value of f(x) as close to 7 as we want by taking x sufficiently close to 1.

7. Using inequalities, Using inequalities, For instance,
In Table 1, if x = 0.9 , f(x) = In Table 2 , if x = 1.1 , f(x) = 7.2
That is , if x is within a distance 0.1 from 1 , f(x) is within a distance 0.2 from 7
Using inequalities,
( 1 )
In Table 1, if x = 0.99 , f(x) = In Table 2 , if x =1 .01 , f(x) = 7.02
That is , if x is within a distance 0.01 from 1, f(x) is within a distance 0.02 from 7
( 2 )
Using inequalities,

8. This shows that we can make the absolute value of the difference
For instance,
In Table 1, if x = , f(x) = In Table 2 , if x = , f(x) = 7.002
That is ,
if x is within a distance from 1 , f(x) is within a distance from 7
Using inequalities,
( 3 )
This shows that we can make the absolute value of the difference
between f(x) and 7 as small as we want by making the absolute value
of the difference between x and 1 small enough.
This situation can be made formally by using the Greek letters
(epsilon) and
(delta) for these small differences.

9. Hence, we say that for any given positive number , there is a positive
number such that
( 4 )
In (1)
In (2)
In (3)
Note that is chosen first and the value of depends on Here, and (4) is equivalent to

10. Graphically, this means that if x lies in the open interval ,
then f(x) lies in the open interval

11. Thus, we have the following formal definition of limit
Or equivalently,
For every
there exists

13. LIMIT RULES
The formal definition of limit does not give us a way to evaluate the limit of a function but it enables us to verify that the limit is correct. To evaluate limits, we have the following limit rules whose proofs are based on the formal definition of limit.

14. Rule 1 : If f(x) = x ( the identity function ) , then
for any a
Rule 2 : If f(x) = k ( the constant function ) , then
for any a
If and where L and M are real
numbers, then
Rule 3 : (Sum/difference Rule)
Rule 4 : (Product Rule)

15. Rule 9 : Limit of a Rational Function
Rule 5 : ( Constant multiple )
Rule 6 : (Quotient rule)
Rule 7 : (Power rule)
The above rules give two immediate results for evaluating limit of a polynomial and a rational function.
Rule 8 : Limit of a Polynomial
If P(x) is a polynomial function, then
Rule 9 : Limit of a Rational Function
If P(x) and Q(x) are polynomials, then
provided Q(a)

16. Examples

17. ONE-SIDED LIMIT
All limit rules for two-sided limit apply for
one-sided limit.
(right hand side limit only)

18. equal Graphically, it means as the graph of f(x).
equal
Graphically, it means as the graph of f(x)
approaches the same point ( 2, 0 ) as when

19. ( 2, 0 )

20. Not equal

21. Graphically, it means as the graph of f(x) does not
approach the same point as when

22. Not equal Graphically, it means as the graph of f(x) does not.
Not equal
Graphically, it means as the graph of f(x) does not
approach the same point as when

23. equal Graphically, it means as the graph of f(x).
equal
Graphically, it means as the graph of f(x)
approaches the same point (3,0) as when

25. Evaluate the limits and function values if they exist from the graph of f sketched below
The domain of f is [0, 5].

26. Consider the function from the left 2 from the right INFINITE LIMIT
Investigate the behavior of f(x) as x goes very very close to 2.
from the left 2
from the right
Take x close to 2 but less than 2
Take x close to 2 but greater than 2

27. Let by assigning values of x very very close to 2 but less than 2.1
1.5
1.75
1.9
1.99
1.999
1.9999
f(x) -1 -2 -4
-10
-100
-1000
-10,000
In symbol, we write this as
Note: is not a number. It is a symbol used to indicate that a quantity is decreasing or increasing without bound.

28. Let by assigning values of x very very close to 2 but greater than 2.3
2.5
2.25
2.1
2.01
2.001
2.0001
f(x) 1 2 4 10
100
1000
10,000
In symbol, we write this as

29. vertical asymptote

30. vertical asymptotes

31. Theorem : Let n be a positive integer, then
Theorem: (Limit of Sum Involving infinity )

32. Theorem: (Limit of Product Involving infinity )

33. Theorem: (Limit of Quotient Involving infinity )