Math 151
Section 2.3
Limit Laws
Limit Laws If lim f ( x ) and lim g ( x) both exist, then the following hold true.
x!a
x!a
1. lim "$ f ( x) ± g ( x)%' = lim f ( x) ± lim g ( x)
& x!a
x!a #
x!a
2. lim "$ cf ( x)%' = c lim f ( x) for any constant c.
&
x!a #
x!a
3. lim "$ f ( x) g ( x)%' = lim f ( x) ( lim g ( x)
& x!a
x!a #
x!a
" f ( x) % lim f ( x)
f ( x)
$
' = x!a
lim
4.
if lim g ( x) " 0 . If g(x) = 0, try to simplify
.
$
'
x!a g x
x!a
g ( x)
g ( x)
$# ( ) '& lim
x!a
(
)
5. lim ( f ( x)) = lim f ( x)
n
x!a
x!a
n
6. lim n f ( x) = n lim f ( x) . If n is even, then it must be true that lim f ( x) > 0 .
x!a
x!a
x!a
Example: Evaluate the following limits. If a limit does not exist, support your answer by evaluating
the left- and right-hand limits.
A. lim ( x 2 + x +1)
5
x!"2
B. lim" 16 " x 2
x!4
Math 151
x4 + x2 " 6
x!1 x 4 + 2x + 3
C. lim
x 2 " x "12
x!"3
x+3
D. lim
E. lim
x!9
F. lim
h!0
G. lim
x!2
x 2 "81
x "3
3+ h " 3
h
x"4
( x " 2)
2
Math 151
# 1
2 &(
((
H. lim %%
" 2
x!1 %
$ x "1 x "1('
I. lim
x!4
x"4
x"4
# x + 3 &(
%
((
J. lim %% 2
x!"3% x + 3x (
('
%$
#
&
%% 1 1 ((
K. lim" % " ((
x!0 %
%$ x x ('
Math 151
#
%
4 + 5x if x < !1
%
%
%x
if !1" x <1
L. lim f ( x) and lim f ( x ) for f ( x) = %
$
x!"1
x!1
%
3
if x = 1
%
%
%
%
&4 ! x if x >1
The Squeeze Theorem If f ( x) ! g ( x) ! h( x) for all x in an open interval containing x = a (except
possibly at a) and lim f ( x) = lim h( x) = L then lim g ( x) = L.
x!a
x!a
x!a
Example: Given 3x ! f ( x) ! x 3 + 2 for all x in the interval (0, 3), find lim f ( x).
x!1
" 1%
Example: Find lim x 4 sin $$ '''.
$# x '&
x!0