Section 2.3 - Calculating Limits Using
The Limit Laws
Introduction
Calculating limits numerically or graphically can be tedious, there must be a better way. Given certain conditions,
there are shortcuts that allow us to easily evaluate limits.
Limits Laws
We have shortcuts that allow us to quickly find the limit of certain classes of functions, such as polynomials and
rational functions.
Limit Laws
Suppose that c is a constant and the limits limxØa f HxL and limxØa gHxL exist. Then
1.
2.
3.
4.
5.
limxØa @ f HxL + gHxLD = limxØa f HxL + limxØa gHxL
limxØa @ f HxL - gHxLD = limxØa f HxL - limxØa gHxL
limxØa @c f HxLD = c limxØa f HxL
limxØa @ f HxL ÿ gHxLD = limxØa f HxL ÿ limxØa gHxL
limxØa B
f HxL
gHxL
F=
limxØa f HxL
limxØa gHxL
if limxØa gHxL ∫ 0
6.
limxØa @ f HxLDn = @limxØa f HxLDn
7.
limxØa c = c
8.
limxØa x = a
9.
limxØa xn = an where n is a positive integer
10.
limxØa
n
x =
n
a
where n is a positive integer
(If n is even, we assume that a > 0.
11.
limxØa
n
f HxL =
n
limxØa f HxL
where n is a positive integer
[If n is even then we assume that limxØa f HxL > 0.]
2
Lecture_02_03.nb
Example:
step.
Use the limit laws to evaluate the limit. Indicate the limit law(s) used at each
limxØ3 I2 x3 - x2 + 7 x - 8M
Direct Substitution Property
If f is a polynomial or a rational function and a is in the domain of f, then limxØa f HxL = f HaL.
Example: Use the Direct Substitution Property to Evaluate the limits.
a.
limxØ-3 I5 x2 - 2 x + 8M
b.
limxØ4
x2 + 1
2x-3
In some cases, we cannot evaluate the limit directly. However, a couple of theorems will allow us to get around this
problem.
If f HxL = gHxL when x ∫ a, then limxØa f HxL = limxØa gHxL, provided the limits exist.
Example: Evaluate the limit.
limxØ2
x2 - 4
x-2
Lecture_02_03.nb
Theorem 1
limxØa f HxL = L if and only if limxØa- f HxL = L = limxØa+ f HxL
Example: Evaluate the limit if f HxL =
limxØ-1 f HxL
:
1-2x
if x < -1
x -2
if x ¥ -1
2
.
Theorem 2
If f HxL § gHxL when x is near a (except possibly at a) and the limits of f and g both exist as x approaches a,
then
limxØa f HxL § limxØa gHxL
Theorem 3 - The Squeeze Theorem
If f HxL § gHxL § hHxL when x is near a (except possibly at a) and
limxØa f HxL = limxØa hHxL = L
then limxØa gHxL = L .
Example: Evaluate the limit.
limxØ0 x2 sin
ü
1
x
3