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Chapter Two
LIMITS AND CONTINUTY
The most basic use of limits is to describe how a function behavior as the
independent variable approaches a given value.
DEFINITION:
If the value of f(x) can be made as close as we like to L by taking the value
of x sufficiently close to a (but not equal a), then we write:
lim f ( x)  L
xa
Which is read "the limit of f(x) as x approaches a is L".
Properties of limits:
1. If f(x) = k,
2. If
then
lim f1 ( x)  L1 and
xa
lim f ( x)  k
where a and k are real numbers.
xa
lim f 2 ( x)  L2 , then:
xa
(a) Sum rule:
lim [ f1 ( x)  f 2 ( x)]  L1  L2
(b) Difference rule:
lim [ f1 ( x)  f 2 ( x)]  L1  L2
(c) Product rule:
lim [ f1 ( x). f 2 ( x)]  L1.L2
(d) Constant multiple rule:
lim k. f1 ( x)  k.L1
(e) Quotient rule:
lim f 1 ( x)  L1 ;
x a
2
2
xa
xa
xa
xa
f ( x)
L
(where k is constant)
L2  0
2
r
[ f1 ( x)] s  L1
lim
xa
(f) Power rule:
r
s
(if s is even number L1 > 0)
2
n
2
n
lim (c0  c1 x c 2 x .....cn x )  c0  c1a c 2 a .....cn a
3. Polynomial
xa
sin x
1
4. lim
x
x0
5. Sandwich theorem:
If g ( x)  f ( x)  h( x) are three functions such that:
lim g ( x)  lim h( x)  L
xa
xa
then lim f ( x)  L .
xa
Note:
1. For sake of convenience in dealing with indeterminate forms, we define
the following arithmetic operations with real numbers, +∞ and -∞. Let c
be a real number and c > 0. Then we define:
+∞ +∞= +∞, -∞ -∞ = -∞, c(+∞) = +∞, c(-∞) = -∞, (-c)( +∞)=-∞, (-c)( -∞) = +∞,
c
c
c
c
 0,
 0,
 0,
 0 , () c   , () c  0 , (+∞) (+∞) = +∞,




(+∞) (-∞) = -∞, (-∞)(-∞) = +∞
2. The following operations are indeterminate quantities:
0
0
( ,

,

,
0* )
Right-hand limits and left-hand limits
The notation for the right-hand limit is
lim f ( x)
xc
"The limit of f(x) as x approaches c from the right"
The (+) is there to say that x
approaches c through values
greater than c on the line
numbers.
-∞
c
x
From right
+∞
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The notation for the left-hand limit is
lim f ( x)
x c 
"The limit of f(x) as x approaches c from the left"
The (-) is there to say that x
approaches c through values
-∞
less than c on the line
x
c
+∞
From left
numbers.
Limit Involving Infinity:
It means that the limits include x   or x   and
limf(x)=∞
or
limf(x)=-∞
Let y 
1
then
x
1
1. lim  
x0 x
the limit does not exit.

1
2. lim  
x0 x
the limit does not exit.

1
3. lim  0
x x
1
4. lim  0
x x
Continuous Functions:
DEFINITION:
- Continuity at interior points:
A function y=f(x) is continuous at an interior point c of its domain if:
lim f ( x)  f (c)
xc
- Continuity at end-points:
+∞
+∞
a
c
A function y= f (x) is continuous at a left end-point a of its domain if:
lim f ( x)  f (a)
xa
A function y=f(x) is continuous at a right end-point b of its domain if:
lim f ( x)  f (b)
xb
b
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Continuous Functions:
A function is continuous if it is continuous at each point of its domain.
Discontinuity at a point:
If a function f (x) is not continuous at a point c, we say that f (x) is
discontinuous at c and call c a point of discontinuity of f (x).
The Continuity Test
The function y=f(x) is continuous at x=c if and only if the following
statements are true:1. f (c) exists (c lies in the domain of f).
2. lim f ( x) exists (f has a limit as x→c).
x c
3. lim f ( x)  f (c) (the limit equals the function value).
xc