Answers and Explanations 199
201.
− 2
Begin by using a trigonometric identity in the numerator and then factor:
lim
cos
sin
cos
lim cos
sin
sin
cos
lim
(
x
x
x
x
x
x
x
x
x
x
→
→
→
−
=
−
−
=
π
π
π
4
4
2
2
4
2
c cos
sin )(cos
sin )
sin
cos
x
x
x
x
x
x
−
+
−
Next, factor out a –1 from the numerator and simplify:
lim
(cos
sin )(cos
sin )
sin
cos
lim
( )( cos
s
x
x
x
x
x
x
x
x
x
→
→
−
+
−
=
− −
+
π
π
4
4
1
i in ) (cos
sin )
sin
cos
lim
( )(sin
cos )(cos
si
x
x
x
x
x
x
x
x
x
[
]
+
−
=
−
−
+
→ π
4
1
n n )
sin
cos
lim( )(cos
sin )
( ) cos
sin
(
x
x
x
x
x
x
−
=
−
+
= −
+
(
)
= −
→ π
π
π
4
1
1
4
4
1) ) 2
2
2
2
2
+






= −
202.
5
9
You want to rewrite the expression so you can use lim sin
x
x
x
→
=
0
1. Begin by multiplying
the numerator and denominator by 1
x
and rewrite the expression as the product of two
fractions:
lim
sin( )
sin( )
lim
sin( )
sin( )
lim
sin( )
x
x
x
x
x
x
x
x
x
x
x
→
→
→
=
=
0
0
0
5
9
5
9
5 ⋅⋅






x
x
sin( )
9
Answers
201–300