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1.3 EVALUATING LIMITSANALYTICALLY
Direct Substitution• If the the value of c is contained in the domain (the
function exists at c) then
limx cf (x) f (c)
Direct Substitution is valid for ALL polynomials and rational functions with non-zero denominators
1) Find
2) Find
94lim 24
3
xx
x
limx 3
x 3
x 2
Properties Of Limits
Basic - let b and c be real numbers and n be a positive integer
I. Constant
II. Identity
III. Power
limx cb
b
limx cx n
limx cx
c
c n
Properties Of LimitsLet L, K, b, and, c be real numbers, let n be a
positive integer, and
1. Scalar Multiple:
2. Sum/Difference:
limx cf (x) L and lim
x cg(x) K
limx c
b f (x) L
limx
3
2
2 x 1
2
2lim
x3
2
x 1
2
2(1)
limx c
f (x) g(x) LK
limx 4
3x 2 2x limx 4
3x 2 limx 4
2x
3(4)2 2(4)
Properties Of LimitsLet L, K, b, and, c be real numbers, let n be a
positive integer, and
3. Power:
4. Product:
limx cf (x) L and lim
x cg(x) K
limx c
f (x) g(x) L K
limx 0
2x 1 x 3 limx 0
2x 1 limx 0
x 3
2(0) 1 0 3
limx c
f (x) ab L
ab
Properties Of LimitsLet L, K, b, and, c be real numbers, let n be a
positive integer, and
5. Quotient:
limx cf (x) L and lim
x cg(x) K
limx c
f (x)
g(x)
L
K
K 0
limx 4
x 2 9
x
limx 4
x 2 9
limx 4x
(4)2 9
4
3) Find
limx1
x 2 x 2
x 2 1 0
0
Technique 1: Rewrite the function by factoring out Common factors
4) Find
limx 3
x 1 2
x 3 0
0
Technique 2: Rationalize the numeratorBy multiplying by the complex conjugate
5) Find
limx 0
1
2 x
1
2x 0
0
Technique 3: Use algebra to rewrite thethe 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
6) Find
7) Find
limx 2
x 2 3x 10
x 2 x 6
limx 0
2(x x) 2x
x
8) Find
limx 0
x 1 1
x
9) Use and2)(lim
xfcx
3)(lim
xgcx
)](5[lim xgcx
)]()([lim xgxfcx
)]()([lim xgxfcx
)(
)(lim
xg
xfcx
Homework
Page 67
# 5 – 25 odd, 37, 38, 39, 41-57 odd,
