MATH 1910
Section 2.3 Calculating Limits Using the Limit Laws
Look at the limit laws beginning on page 111 of the textbook. An important note is that the limit laws can only
be applied to functions whose limits EXIST.
Ex. Given that
lim ( )
xa
fx
→
=2 lim ( )
xa
gx
→
=
−
4 lim ( )
xa
hx
→
=
1,
evaluate the following limits, if they exist.
a) lim[ () ()]
xa
fx hx
→
−3 b) lim[ ( )]
xa
gx
→
2 c) lim ( )
()
()
xa
gx
hx fx
→
+
d)
lim ()
xa
fx
→
1 e)
lim ( ) ( )
xa
hx gx
→
−3
3
Ex. Evaluate the following limits, if they exist, without the aid of a calculator:
a)
lim
x
x
x
x
→−
−
+−
12
2
43
b) lim
x
xx
→−
++
1
327
There are times when you have to do some algebraic maneuvering such as factoring both sides of a rational
expression, rationalizing the NUMERATOR of a rational expression before you can evaluate a limit.
Ex. Evaluate the following limits, if they exist, without the aid of a calculator:
a) lim
x
xx
x
→−
−−
+
3
212
3 b) lim
x
x
x
→
−
−
1
2
3
1
1 c) lim
h
h
h
→
+−
0
55
=−+
+
→−
lim ()()
x
xx
x
3
43
3 =
lim ()()
()( )
x
xx
xxx
→
+
−
−++
12
11
11
=
lim
h
h
h
h
h
→
+− ⋅++
++
0
5555
55
cancel out the ( )
x
+3 = lim ()
()
x
x
xx
→
+
++
12
1
1 rationalize the numerator
=−
→−
lim ( )
x
x
3
4= – 7 = 2
3 =
lim ()
h
h
hh
→
+−
⋅++
0
55
55
=
lim ()
h
h
hh
→⋅++
055
=
lim
hh
→++
=
0
1
55
1
25
Ex. Evaluate the one sided limit lim
xxx
→+
−
L
N
MO
Q
P
0
11
using your knowledge of the absolute value function.
