Page 1 | © 2012 by Janice L. Epstein 2.3 Limit Laws Page 2 | © 2012 by Janice L. Epstein 2.3 Limit Laws Calculating Limits Using the Limit Laws (2.3)
EXAMPLE 1
Given that lim f ( x) = 3 , find the following using the limit laws.
Suppose that c is a constant, n is a positive integer, and the limits
lim f ( x) and lim g ( x)
x a
x a
exist. Then the following laws can be used to calculate limits.
Note that n is a positive integer
1.
lim [ f ( x) + g ( x) ] = lim f ( x) + lim g ( x)
2.
lim [ f ( x ) - g ( x) ] = lim f ( x ) - lim g ( x)
3.
lim [cf ( x) ] = c lim f ( x )
4.
lim [ f ( x) ⋅ g ( x) ] = lim f ( x ) ⋅ lim g ( x )
5.
lim
x a
x a
x a
x a
x a
f ( x) + 6
x 2 2 f ( x ) + x + 11
b) lim
x a
x a
x a
f ( x)
f ( x) lim
= x a
g ( x ) lim g ( x)
2
a) lim éê( f ( x)) - 2 x + 4ùú
x 2 ë
û
x a
x a
x a
x 2
x a
if lim g ( x) ¹ 0
x a
x a
6.
n
lim [ f ( x ) ] = éê lim f ( x )ùú
x a
ë x a
û
7.
lim c = c
8.
lim x = x
9.
lim x n = a n
n
EXAMPLE 2
Find the following limits, if they exist
x a
a) lim ( x 2 + x + 1)
5
x-2
x a
x a
10. lim n x = n a (if n is even, assume a > 0 )
x a
11. lim n f ( x) = n lim f ( x) (if n is even, assume lim f ( x) > 0 )
x a
a¥
x a
Page 3 | © 2012 by Janice L. Epstein 2.3 Limit Laws Page 4 | © 2012 by Janice L. Epstein x 2 - x -12
x-3
x +3
b) lim
c) lim
x64
(
3
x +3 x
)
x2 - x - 3
d) lim
x-1
x +1
(2 + h) - 8
3
e) lim
h0
h
f) lim+
x1.5
(
3 + 2x + x
)
x +1
x 4 ( x - 4) 2
g) lim
t 3 - t 2 - t + 10
h) lim 2
t -2
t + 3t + 2
2.3 Limit Laws Page 5 | © 2012 by Janice L. Epstein 2.3 Limit Laws Page 6 | © 2012 by Janice L. Epstein 2.3 Limit Laws Some special functions are
ìï x if x ³ 0
Absolute value = x = ï
í
ïïî-x if x < 0
if x < 0
ïìï x
ï 2
c) lim f ( x) and lim f ( x) given that f ( x ) = í x
if 0 < x £ 2
x 0
x 2
ïï
ïïî4 - x if x > 2
Greatest Integer =  x 
= the largest integer that £ to x
EXAMPLE 3
Find the following limits, if they exist.
a) lim
x-4
x+4
x+4
b) lim  x 
x-2
EXAMPLE 4
The position of a moving particle at time t, 0 £ t < 5 is given by
2t -10
t -5
. What is the anticipated position of
, 2
r (t ) =
t - 5 t - 4t - 5
the particle at t = 5 ?
Page 7 | © 2012 by Janice L. Epstein 2.3 Limit Laws Page 8 | © 2012 by Janice L. Epstein Theorem:
If f ( x) £ g ( x) for all x in an open interval that contains a (except
possibly at a ) and the limits of f and g both exist as x approaches
a, then lim f ( x) £ lim g ( x)
EXAMPLE 6
x a
x a
1
g ( x ) = x sin
x
0.4
f ( x) = x
h( x) = - x
-0.4
0.4
-0.4
1
To find lim x sin , we can use the Squeeze Theorem.
x 0
x
If f ( x) £ g ( x) £ h( x) for all x in an open interval that contains a
(except possibly at a) and lim f ( x) = L = lim h( x) then
lim g ( x) = L
x a
x a
x a
lim x sin
x 0
1
=
x
EXAMPLE 5
Given that 3 x £ f ( x ) £ x 3 + 2 for all the x in the interval (0, 3),
find lim f ( x)
x1
2.3 Limit Laws 3 x 2 + ax + a + 3
x-2
x2 + x - 2
exists? If so, what is the value of the limit?
Is there a value or values of a, such that lim