This document discusses limit laws and how to evaluate limits. It provides 6 limit laws for evaluating limits of functions, including laws for sums, products, quotients, roots, and composite functions. It also includes an example of applying these laws to evaluate several example limits. Finally, it discusses the squeeze theorem for evaluating limits and provides an example of applying that theorem.

1. c Amy Austin, September 16, 2015 1
Section 2.3: Limit Laws
Limit Laws If lim
x→a
f(x) and lim
x→a
g(x) both exist and c is any constant. Then:
(1) lim
x→a
(f(x) ± g(x)) = lim
x→a
f(x) ± lim
x→a
g(x)
(2) lim
x→a
cf(x) = c lim
x→a
f(x)
(3) lim
x→a
f(x)g(x) = lim
x→a
f(x) lim
x→a
g(x)
(4) lim
x→a
f(x)
g(x)
=
lim
x→a
f(x)
lim
x→a
g(x)
, provided lim
x→a
g(x) 6= 0. If lim
x→a
g(x) = 0, simplify
f(x)
g(x)
(5) lim
x→a
(f(x))n
=
lim
x→a
f(x)
n
(6) lim
x→a
n
q
f(x) = n
q
lim
x→a
f(x). In the event n is even, we must have lim
x→a
f(x) 0.
EXAMPLE 1: If it is known that lim
x→2
f(x) = 3, find lim
x→2
q
(f(x))2 + 6
f(x) + 2x + 11
EXAMPLE 2: Find the limit. If the limit does not exist, explain why.
(a) lim
x→1
x4
+ x2
− 6
x4 + 2x + 3
(b) lim
x→−3
x2
− x − 12
x + 3

2. c Amy Austin, September 16, 2015 2
(c) lim
x→9
x2
− 81
√
x − 3
(d) lim
x→6
1
6
−
1
x
x − 6
(e) Find lim
x→5
f(x), lim
x→1
f(x), and lim
x→−1
f(x) where f(x) =







4 + 5x if x −1
x if −1 ≤ x 1
3 if x = 1
4 − x if x 1
.

3. c Amy Austin, September 16, 2015 3
(f) Find lim
x→2−
|x − 2|
x − 2
(g) Let f(x) =
x2
+ 3x
|x + 3|
. Find lim
x→−3−
f(x) and lim
x→−3+
f(x).
Squeeze (aka sandwich) Theorem If f(x) ≤ g(x) ≤ h(x) for all x in an interval
containing x = a (except possibly at x = a) and lim
x→a
f(x) = lim
x→a
h(x) = L, then
lim
x→a
g(x) = L.
EXAMPLE 3: Given 3x ≤ f(x) ≤ x3
+2 for all x in the interval (0, 3), find lim
x→1
f(x).