2.2b
Section 2.2b: Limit Laws and the Sandwich/Squeeze Theorem
Theorem 1: Limit Laws
such that gcd(r , s )  1 and lim f ( x)  L and lim g ( x)  M then:
If L, M , c, k  , r , s 
x c
1. Sum Rule: lim( f ( x)  g ( x)) 
x c
2. Difference Rule: lim( f ( x)  g ( x)) 
x c
3. Product Rule: lim( f ( x)  g ( x)) 
x c
4. Constant Multiple Rule: lim(k  f ( x)) 
x c
5. Quotient Rule: lim
x c
f ( x)

g ( x)
6. Power Rule: lim( f ( x))
xc
Example:
a) lim x5  4 x2  x  10 
xc
x3  2

x 2 5 x  x 2
b) lim
r
s

x c
1
2.2b
c) lim( x 4  7)
x1
5
3

Theorem 2: Limits of Polynomials
If P( x)  an x n  an1x n1 
 a0 then lim P ( x) 
x c
Theorem 3: Limits of Rational Functions:
If P( x), Q( x) are polynomials with Q (c )  0 , then lim
xc
Example:
x3  4 x 2  3
Find lim
x1
x2  5
What about zero denominators??
Example: Consider lim
x5
x2  6 x  5
x 5
P( x)

Q( x)
2
2.2b
Why can’t I always use a computer or calculator?
 
Consider limsin  
x 0
x
1.0
0.5
2
Let’s make a table:
1
1
0.5
1.0
 
sin  
x
x
-.001
.001
-.0001
.0001
Example (in text Example 9)
lim
x0
x 2  100  10
x2
2
3
2.2b
4
Theorem 4: Sandwich/Squeeze Theorem:
Suppose that g ( x)  f ( x)  h( x) for all x in some open interval containing a point c, except
possibly at x=c. Suppose also that lim g ( x)  lim h( x)  L . Then,
xc
xc
lim f ( x)  L
x c
Example:
x2
x2
Suppose we know that 1   u ( x)  1 
for all x  0 . Find lim u ( x) .
x 0
4
2
Theorem 5:
Suppose that f ( x)  g ( x) for all x in some open interval containing a point c, except possibly at
x=c, and the limits of f and g both exist as x approaches c. Then, lim f ( x)  lim g ( x)
x c
Example: Is lim x3  lim x 2 for c  (1, 4) ?
xc
xc
x c