Download sect. 1-3

1.3 EVALUATING LIMITS
ANALYTICALLY
Direct Substitution
• If the the value of c is contained in the domain (the
function exists at c) then
lim f (x)  f (c)
x c
Direct Substitution is valid for ALL polynomials and
rational functions with non-zero denominators

1) Find
2)

lim x  4 x  9
4
x3
x3
Find lim
x 3 x  2
2
Properties Of Limits
Basic - let b and c be real numbers and n be a
positive integer
I.
Constant
lim b  b
x c
II.
III.
Identity

Power



lim x  c
x c
n
lim x  c n
x c

Properties Of Limits
Let L, K, b, and, c be real numbers, let n be a
positive integer, and lim f (x)  L and lim g(x)  K
x c
1.
b  f (x)  L

Scalar Multiple: lim
x c
  1 
 1 
lim 2x  
 2lim x  

3
2  x  3 2 
x   
2
2.
x c
2
 2(1)

f (x)  g(x)  L  K

Sum/Difference:lim
x c
lim 3x 2  2x  lim 3x 2  lim 2x
x 4
x 4
x 4
2

 3(4)  2(4)

Properties Of Limits
Let L, K, b, and, c be real numbers, let n be a
positive integer, and lim f (x)  L and lim g(x)  K
x c
3.
a
a
Power: lim  f (x)  L
b
x c
b
x c

4.
f (x)  g(x)  L  K

Product: lim
x c
lim 2x 1x  3 lim 2x 1lim x  3
x0

x0
x0
 2(0)  10  3

Properties Of Limits
Let L, K, b, and, c be real numbers, let n be a
f (x)  L and lim g(x)  K
positive integer, and lim
x c
x c
f (x)  L

K 0
5. Quotient: lim 

x cg(x)  K

 x 2  9  lim x 2  9
 x 4
lim 
x 4
x

 x 
 lim
x 4

2
(4)  9

4
3)
Find
x2  x  2
lim
2
x 1
x 1
0

0

Technique 1: Rewrite the function by
factoring out Common factors
4)
Find
lim
x 3
x 1  2
x3
0

0

Technique 2: Rationalize the numerator
By multiplying by the complex conjugate
5)
Find
1
1

lim 2  x 2
x 0
x
0

0

Technique 3: Use algebra to rewrite the
the function
Strategies for Limits
1) Determine by recognition whether a limit
can be evaluated by direct substitution
2) If direct substitution fails, try to use some
technique (cancellation, rationalization,
or algebraic manipulation)
3) Use a graph or table to verify your
conclusion
x 2  3x 10
6) Find lim
x 2 x 2  x  6

2(x  x)  2x
7) Find lim
x 0
x

8) Find
lim
x 0

x  1 1
x
9) Use
lim f ( x )  2and lim g ( x )  3
x c
lim [5 g ( x)] 
x c
lim [ f ( x)  g ( x)] 
x c
lim [ f ( x) g ( x)] 
x c
f ( x)
lim

x c g ( x )
xc
Homework
Page 67
# 5 – 25 odd, 37, 38, 39, 41-57 odd,