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
limb  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.
x c
b f (x)  L

Scalar Multiple: lim
xc
  1 
 1 
lim2x  
 2lim3x  
3 
2  x   2 
x
2
2
  2(1)
2. Sum/Difference:lim f (x)  g(x)  L  K
xc
lim3x 2  2x lim 3x 2  lim2x
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
xc
3.
a
a
Power: lim f (x)  L
b
b
x c

4. 
Product:
lim f (x) g(x)  L K
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  1  2
x 3
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

x  1 1
lim
x 0
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,